| L(s) = 1 | + 2-s − 2·4-s − 6·7-s − 3·8-s + 6·11-s − 3·13-s − 6·14-s + 16-s − 4·17-s − 5·19-s + 6·22-s − 2·23-s − 3·26-s + 12·28-s − 31-s + 2·32-s − 4·34-s − 6·37-s − 5·38-s + 6·41-s − 18·43-s − 12·44-s − 2·46-s + 47-s + 13·49-s + 6·52-s − 7·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 2.26·7-s − 1.06·8-s + 1.80·11-s − 0.832·13-s − 1.60·14-s + 1/4·16-s − 0.970·17-s − 1.14·19-s + 1.27·22-s − 0.417·23-s − 0.588·26-s + 2.26·28-s − 0.179·31-s + 0.353·32-s − 0.685·34-s − 0.986·37-s − 0.811·38-s + 0.937·41-s − 2.74·43-s − 1.80·44-s − 0.294·46-s + 0.145·47-s + 13/7·49-s + 0.832·52-s − 0.961·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 163 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 21 T + 233 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 137 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 203 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.364548413281520664964006656047, −9.331817130203183434841310845340, −8.892349457328620155463952184072, −8.767499683874449884153001477285, −7.982644500631273399449482634357, −7.47221989044265891151762632392, −6.83618840359182586655778819742, −6.52463530889994092956135692081, −6.20349528195766851787401906927, −6.11738561864168487750830607320, −5.06964172214320127073450208450, −4.89066442932451777174666070280, −4.21698076912348442642115759196, −3.99124950656939240773113677567, −3.28553482112215943120812418959, −3.27560596520457528117576023435, −2.30691890247733008421129852217, −1.53511278705354164363266331654, 0, 0,
1.53511278705354164363266331654, 2.30691890247733008421129852217, 3.27560596520457528117576023435, 3.28553482112215943120812418959, 3.99124950656939240773113677567, 4.21698076912348442642115759196, 4.89066442932451777174666070280, 5.06964172214320127073450208450, 6.11738561864168487750830607320, 6.20349528195766851787401906927, 6.52463530889994092956135692081, 6.83618840359182586655778819742, 7.47221989044265891151762632392, 7.982644500631273399449482634357, 8.767499683874449884153001477285, 8.892349457328620155463952184072, 9.331817130203183434841310845340, 9.364548413281520664964006656047
