| L(s) = 1 | + 2·2-s + 15·4-s + 20·5-s + 50·8-s + 40·10-s − 20·11-s + 208·13-s + 132·16-s + 116·17-s − 192·19-s + 300·20-s − 40·22-s + 28·23-s + 252·25-s + 416·26-s − 592·29-s + 104·31-s + 510·32-s + 232·34-s + 248·37-s − 384·38-s + 1.00e3·40-s − 40·41-s − 1.44e3·43-s − 300·44-s + 56·46-s − 96·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 15/8·4-s + 1.78·5-s + 2.20·8-s + 1.26·10-s − 0.548·11-s + 4.43·13-s + 2.06·16-s + 1.65·17-s − 2.31·19-s + 3.35·20-s − 0.387·22-s + 0.253·23-s + 2.01·25-s + 3.13·26-s − 3.79·29-s + 0.602·31-s + 2.81·32-s + 1.17·34-s + 1.10·37-s − 1.63·38-s + 3.95·40-s − 0.152·41-s − 5.10·43-s − 1.02·44-s + 0.179·46-s − 0.297·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(20.14636082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.14636082\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T - 11 T^{2} + p T^{3} + 129 T^{4} + p^{4} T^{5} - 11 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 4 p T + 148 T^{2} - 8 p T^{3} - 2121 T^{4} - 8 p^{4} T^{5} + 148 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 20 T - 10 p^{2} T^{2} - 21040 T^{3} + 288139 T^{4} - 21040 p^{3} T^{5} - 10 p^{8} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8 p T + 5848 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 116 T + 4316 T^{2} + 79576 T^{3} - 6707297 T^{4} + 79576 p^{3} T^{5} + 4316 p^{6} T^{6} - 116 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 192 T + 14898 T^{2} + 1583616 T^{3} + 182609099 T^{4} + 1583616 p^{3} T^{5} + 14898 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 28 T - 22178 T^{2} + 38416 T^{3} + 369678627 T^{4} + 38416 p^{3} T^{5} - 22178 p^{6} T^{6} - 28 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 296 T + 62994 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 104 T - 46470 T^{2} + 238784 T^{3} + 2071962659 T^{4} + 238784 p^{3} T^{5} - 46470 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 248 T - 50570 T^{2} - 2670464 T^{3} + 6879492955 T^{4} - 2670464 p^{3} T^{5} - 50570 p^{6} T^{6} - 248 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 20 T + 42020 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 720 T + 268614 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 96 T - 75734 T^{2} - 11778816 T^{3} - 4519545597 T^{4} - 11778816 p^{3} T^{5} - 75734 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 268 T + 15314 T^{2} + 64653392 T^{3} - 29663922677 T^{4} + 64653392 p^{3} T^{5} + 15314 p^{6} T^{6} - 268 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 616 T - 24014 T^{2} - 4489408 T^{3} + 42675213435 T^{4} - 4489408 p^{3} T^{5} - 24014 p^{6} T^{6} + 616 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 16 T - 453752 T^{2} + 736 T^{3} + 154544782567 T^{4} + 736 p^{3} T^{5} - 453752 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 144 T - 36822 T^{2} + 78331392 T^{3} - 93382080373 T^{4} + 78331392 p^{3} T^{5} - 36822 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 988 T + 852210 T^{2} + 988 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 104 T - 448320 T^{2} - 33165392 T^{3} + 55264032335 T^{4} - 33165392 p^{3} T^{5} - 448320 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 944 T - 206334 T^{2} - 105154048 T^{3} + 521988143075 T^{4} - 105154048 p^{3} T^{5} - 206334 p^{6} T^{6} - 944 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 1016 T + 1388838 T^{2} + 1016 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 388 T - 1067188 T^{2} - 74575928 T^{3} + 879761079727 T^{4} - 74575928 p^{3} T^{5} - 1067188 p^{6} T^{6} + 388 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 488 T + 1167280 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62896441071531748233403601459, −7.30633880186448245478213409996, −6.74750940058676248173332926419, −6.74472369227875786893920202009, −6.56153465203783585694929606169, −6.32181010519490488642911067889, −6.26380608404244140987771138746, −5.66767602592171733384019376857, −5.63772738394136881441678953044, −5.53003537198711305594591695808, −5.42619903762497557994698507791, −4.74015876659192877055192795973, −4.49134761254252384294577137581, −4.07810524328721266507511360819, −3.91719241024404697659632781407, −3.60974801325398666362746762784, −3.18629329963424325766841295994, −2.98664271962340096130436673436, −2.83668587492723389470072500433, −1.93090394129773670482790925844, −1.89353054178358461777811314164, −1.68328282566716467142245798787, −1.47065383913897400930904364619, −1.22430955073623662555331717348, −0.38889937368521281850529144697,
0.38889937368521281850529144697, 1.22430955073623662555331717348, 1.47065383913897400930904364619, 1.68328282566716467142245798787, 1.89353054178358461777811314164, 1.93090394129773670482790925844, 2.83668587492723389470072500433, 2.98664271962340096130436673436, 3.18629329963424325766841295994, 3.60974801325398666362746762784, 3.91719241024404697659632781407, 4.07810524328721266507511360819, 4.49134761254252384294577137581, 4.74015876659192877055192795973, 5.42619903762497557994698507791, 5.53003537198711305594591695808, 5.63772738394136881441678953044, 5.66767602592171733384019376857, 6.26380608404244140987771138746, 6.32181010519490488642911067889, 6.56153465203783585694929606169, 6.74472369227875786893920202009, 6.74750940058676248173332926419, 7.30633880186448245478213409996, 7.62896441071531748233403601459
