| L(s) = 1 | + 9·7-s + 5·13-s − 5·25-s − 18·31-s + 22·37-s − 18·43-s + 47·49-s + 61-s + 27·67-s + 14·73-s + 9·79-s + 45·91-s − 19·97-s + 27·103-s − 4·109-s + 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯ |
| L(s) = 1 | + 3.40·7-s + 1.38·13-s − 25-s − 3.23·31-s + 3.61·37-s − 2.74·43-s + 47/7·49-s + 0.128·61-s + 3.29·67-s + 1.63·73-s + 1.01·79-s + 4.71·91-s − 1.92·97-s + 2.66·103-s − 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.054314132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.054314132\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.718007030682469486275479393183, −9.470570090356143421798942086014, −8.940951847667308329536116191992, −8.328925532816339426799889938216, −8.268224373205767301190406580411, −8.046828685041279380352808564356, −7.40306571360408387840554494653, −7.35954374425345910510747919885, −6.46487655691979086427647405746, −6.09660609737121073298803901806, −5.38647248708788311457938252948, −5.32245723708886382928064509824, −4.83306592173024571555445160727, −4.33669709400533156588235149354, −3.77494864241125368317108564764, −3.59655871859952174450881675602, −2.31679278661667540193135661397, −2.03663130615716882751725528435, −1.50711162235389149707276890328, −0.942113657085336780854598286456,
0.942113657085336780854598286456, 1.50711162235389149707276890328, 2.03663130615716882751725528435, 2.31679278661667540193135661397, 3.59655871859952174450881675602, 3.77494864241125368317108564764, 4.33669709400533156588235149354, 4.83306592173024571555445160727, 5.32245723708886382928064509824, 5.38647248708788311457938252948, 6.09660609737121073298803901806, 6.46487655691979086427647405746, 7.35954374425345910510747919885, 7.40306571360408387840554494653, 8.046828685041279380352808564356, 8.268224373205767301190406580411, 8.328925532816339426799889938216, 8.940951847667308329536116191992, 9.470570090356143421798942086014, 9.718007030682469486275479393183
