| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 6·11-s + 4·13-s − 4·14-s + 5·16-s + 10·17-s − 4·19-s + 12·22-s + 12·23-s + 2·25-s − 8·26-s + 6·28-s + 2·31-s − 6·32-s − 20·34-s + 2·37-s + 8·38-s − 8·41-s + 4·43-s − 18·44-s − 24·46-s + 8·47-s + 49-s − 4·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 1.80·11-s + 1.10·13-s − 1.06·14-s + 5/4·16-s + 2.42·17-s − 0.917·19-s + 2.55·22-s + 2.50·23-s + 2/5·25-s − 1.56·26-s + 1.13·28-s + 0.359·31-s − 1.06·32-s − 3.42·34-s + 0.328·37-s + 1.29·38-s − 1.24·41-s + 0.609·43-s − 2.71·44-s − 3.53·46-s + 1.16·47-s + 1/7·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64738116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64738116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153835112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153835112\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 149 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 79 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 71 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 140 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22 T + 276 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 188 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 211 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 272 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−7.892092198824653189085311876785, −7.86994813606706874583501118610, −7.38092982695257255645828189381, −7.11399693968654131738831987883, −6.95184947752093021030317913608, −6.29345017517326921079363974109, −5.93283341794444826713330790608, −5.48413965927794410733480460863, −5.46362903217079051726161392262, −5.04388472101687217682963669317, −4.42269169166298018891369880646, −4.19131958713097240294888518133, −3.35266260844437622594969916554, −3.20735363672090928232361934746, −2.68227086965637117352917520824, −2.60165568500008787036826595074, −1.62010086539031180816470320116, −1.54372246277588325061309566647, −0.835592154799277896367630077198, −0.58567976993330723604055100019,
0.58567976993330723604055100019, 0.835592154799277896367630077198, 1.54372246277588325061309566647, 1.62010086539031180816470320116, 2.60165568500008787036826595074, 2.68227086965637117352917520824, 3.20735363672090928232361934746, 3.35266260844437622594969916554, 4.19131958713097240294888518133, 4.42269169166298018891369880646, 5.04388472101687217682963669317, 5.46362903217079051726161392262, 5.48413965927794410733480460863, 5.93283341794444826713330790608, 6.29345017517326921079363974109, 6.95184947752093021030317913608, 7.11399693968654131738831987883, 7.38092982695257255645828189381, 7.86994813606706874583501118610, 7.892092198824653189085311876785
