| L(s) = 1 | + 2-s + 4-s + 3·5-s − 3·7-s + 8-s − 9-s + 3·10-s − 9·11-s − 3·13-s − 3·14-s + 16-s − 18-s + 3·20-s − 9·22-s + 4·25-s − 3·26-s − 3·28-s − 3·31-s + 32-s − 9·35-s − 36-s + 3·40-s − 18·43-s − 9·44-s − 3·45-s + 18·47-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.13·7-s + 0.353·8-s − 1/3·9-s + 0.948·10-s − 2.71·11-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 0.235·18-s + 0.670·20-s − 1.91·22-s + 4/5·25-s − 0.588·26-s − 0.566·28-s − 0.538·31-s + 0.176·32-s − 1.52·35-s − 1/6·36-s + 0.474·40-s − 2.74·43-s − 1.35·44-s − 0.447·45-s + 2.62·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
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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
−9.141467364149856822403122330074, −8.523867804037650047954507928768, −7.970134397614120961710402970600, −7.40520898065405655844502637312, −7.06416105786978693562804481624, −6.35140159592196711898105980958, −5.91400342614681451534250468467, −5.44146563153918335127861626859, −5.14604127007658209979990797055, −4.59419224967991563445829539897, −3.55862756012250695283950694402, −2.84603758475914421230951939886, −2.61187419171134344106616527182, −1.90207308866371550183470703252, 0,
1.90207308866371550183470703252, 2.61187419171134344106616527182, 2.84603758475914421230951939886, 3.55862756012250695283950694402, 4.59419224967991563445829539897, 5.14604127007658209979990797055, 5.44146563153918335127861626859, 5.91400342614681451534250468467, 6.35140159592196711898105980958, 7.06416105786978693562804481624, 7.40520898065405655844502637312, 7.970134397614120961710402970600, 8.523867804037650047954507928768, 9.141467364149856822403122330074
