| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 6·7-s − 4·8-s + 3·9-s + 5·11-s − 6·12-s + 13-s + 12·14-s + 5·16-s − 3·17-s − 6·18-s − 19-s + 12·21-s − 10·22-s + 2·23-s + 8·24-s − 2·26-s − 4·27-s − 18·28-s − 4·29-s + 12·31-s − 6·32-s − 10·33-s + 6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 2.26·7-s − 1.41·8-s + 9-s + 1.50·11-s − 1.73·12-s + 0.277·13-s + 3.20·14-s + 5/4·16-s − 0.727·17-s − 1.41·18-s − 0.229·19-s + 2.61·21-s − 2.13·22-s + 0.417·23-s + 1.63·24-s − 0.392·26-s − 0.769·27-s − 3.40·28-s − 0.742·29-s + 2.15·31-s − 1.06·32-s − 1.74·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5374931838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5374931838\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - T - 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14 T + 114 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 20 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 153 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 209 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 184 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.801012976646974313941080866245, −8.658210302192770748515870195519, −7.989208059748049829370859807515, −7.76927652754224681075974381362, −7.05581320459853936945745470228, −6.68084775328972205910860537781, −6.59688351138234078806255829651, −6.43159868266270945459865001299, −6.20640679644333468405607633455, −5.56031834038327881355201901400, −5.09350219116716253209275603337, −4.62826327387235386446208649450, −3.93654045213171685287709162609, −3.71394305704945923841443109738, −3.00072267742670920775548134199, −2.99075122107767620966670821137, −1.85922975171716059055306602683, −1.70482563237124050045064784887, −0.69274100054239107301987565343, −0.48395788617804900106636547994,
0.48395788617804900106636547994, 0.69274100054239107301987565343, 1.70482563237124050045064784887, 1.85922975171716059055306602683, 2.99075122107767620966670821137, 3.00072267742670920775548134199, 3.71394305704945923841443109738, 3.93654045213171685287709162609, 4.62826327387235386446208649450, 5.09350219116716253209275603337, 5.56031834038327881355201901400, 6.20640679644333468405607633455, 6.43159868266270945459865001299, 6.59688351138234078806255829651, 6.68084775328972205910860537781, 7.05581320459853936945745470228, 7.76927652754224681075974381362, 7.989208059748049829370859807515, 8.658210302192770748515870195519, 8.801012976646974313941080866245
