| L(s) = 1 | + 8·2-s + 48·4-s − 12·5-s − 176·7-s + 256·8-s − 96·10-s − 540·11-s − 446·13-s − 1.40e3·14-s + 1.28e3·16-s − 1.56e3·17-s − 1.59e3·19-s − 576·20-s − 4.32e3·22-s − 1.40e3·23-s − 5.46e3·25-s − 3.56e3·26-s − 8.44e3·28-s − 2.34e3·29-s − 776·31-s + 6.14e3·32-s − 1.24e4·34-s + 2.11e3·35-s + 3.52e3·37-s − 1.27e4·38-s − 3.07e3·40-s − 2.78e4·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.214·5-s − 1.35·7-s + 1.41·8-s − 0.303·10-s − 1.34·11-s − 0.731·13-s − 1.91·14-s + 5/4·16-s − 1.30·17-s − 1.01·19-s − 0.321·20-s − 1.90·22-s − 0.553·23-s − 1.74·25-s − 1.03·26-s − 2.03·28-s − 0.516·29-s − 0.145·31-s + 1.06·32-s − 1.85·34-s + 0.291·35-s + 0.423·37-s − 1.43·38-s − 0.303·40-s − 2.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 5611 T^{2} + 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 176 T + 28290 T^{2} + 176 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 540 T + 310330 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 446 T + 60939 p T^{2} + 446 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1560 T + 3172687 T^{2} + 1560 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1592 T + 5183946 T^{2} + 1592 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1404 T + 515726 p T^{2} + 1404 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2340 T + 11651995 T^{2} + 2340 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 776 T - 33605346 T^{2} + 776 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3526 T + 28328583 T^{2} - 3526 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 27840 T + 421901410 T^{2} + 27840 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 23104 T + 371010858 T^{2} - 23104 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 38784 T + 16628018 p T^{2} + 38784 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 45672 T + 1133302570 T^{2} + 45672 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2160 T + 484762198 T^{2} + 2160 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8974 T + 1242523743 T^{2} - 8974 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 71656 T + 2902035498 T^{2} - 71656 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 348 p T + 3665651218 T^{2} + 348 p^{6} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 60322 T + 3938822259 T^{2} - 60322 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 74392 T + 6604781346 T^{2} - 74392 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 29496 T + 4384376038 T^{2} + 29496 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 110976 T + 7596045367 T^{2} - 110976 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5564 T + 16765832070 T^{2} + 5564 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.72211489963745894871337884338, −11.43886082301709929499872310680, −10.80621909342521914366273981368, −10.34891340594956740226008639303, −9.673235757353710410219372033806, −9.504187877399036259734537973588, −8.330604942194062031230846043129, −7.997157419992202146643359081737, −7.32815896406749557084547914643, −6.66501528106118735266684424428, −6.27853496664103112212502566182, −5.80256945690117048447719467331, −4.79542577539561250533998211043, −4.73328113049888289224908985563, −3.53037218545734001505604040719, −3.43730392989104451699244908297, −2.20580180124576025921583023300, −2.14113876080594673109710957396, 0, 0,
2.14113876080594673109710957396, 2.20580180124576025921583023300, 3.43730392989104451699244908297, 3.53037218545734001505604040719, 4.73328113049888289224908985563, 4.79542577539561250533998211043, 5.80256945690117048447719467331, 6.27853496664103112212502566182, 6.66501528106118735266684424428, 7.32815896406749557084547914643, 7.997157419992202146643359081737, 8.330604942194062031230846043129, 9.504187877399036259734537973588, 9.673235757353710410219372033806, 10.34891340594956740226008639303, 10.80621909342521914366273981368, 11.43886082301709929499872310680, 11.72211489963745894871337884338
