L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s + 9-s − 4·10-s − 4·13-s − 4·16-s + 2·18-s + 4·19-s − 4·20-s − 4·23-s − 3·25-s − 8·26-s − 4·29-s − 8·32-s + 2·36-s + 8·38-s − 8·41-s − 2·45-s − 8·46-s − 8·47-s − 2·49-s − 6·50-s − 8·52-s − 8·58-s − 8·64-s + 8·65-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s + 1/3·9-s − 1.26·10-s − 1.10·13-s − 16-s + 0.471·18-s + 0.917·19-s − 0.894·20-s − 0.834·23-s − 3/5·25-s − 1.56·26-s − 0.742·29-s − 1.41·32-s + 1/3·36-s + 1.29·38-s − 1.24·41-s − 0.298·45-s − 1.17·46-s − 1.16·47-s − 2/7·49-s − 0.848·50-s − 1.10·52-s − 1.05·58-s − 64-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
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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
−8.901300562843048006547859642897, −8.359807263409020964794326620974, −7.75322910345383070042157225257, −7.45215375505830205812920875571, −6.98015762967909310650978534920, −6.37723516419377311680704168609, −5.87305553162616800818494994578, −5.21633161305066184904706219518, −4.88408419403844195396465127791, −4.32723375679961882128708392617, −3.68761035801875766224811205411, −3.39937352297480196255186845413, −2.56794790777344149528973664834, −1.78116417252197338096719161187, 0,
1.78116417252197338096719161187, 2.56794790777344149528973664834, 3.39937352297480196255186845413, 3.68761035801875766224811205411, 4.32723375679961882128708392617, 4.88408419403844195396465127791, 5.21633161305066184904706219518, 5.87305553162616800818494994578, 6.37723516419377311680704168609, 6.98015762967909310650978534920, 7.45215375505830205812920875571, 7.75322910345383070042157225257, 8.359807263409020964794326620974, 8.901300562843048006547859642897