L(s) = 1 | + 0.863·2-s − 31.2·4-s + 44.4·5-s − 36.7·7-s − 54.6·8-s + 38.4·10-s + 302.·11-s − 373.·13-s − 31.7·14-s + 952.·16-s − 280.·17-s + 1.37e3·19-s − 1.39e3·20-s + 261.·22-s − 1.86e3·23-s − 1.14e3·25-s − 322.·26-s + 1.14e3·28-s + 841·29-s + 1.47e3·31-s + 2.57e3·32-s − 242.·34-s − 1.63e3·35-s − 1.17e4·37-s + 1.18e3·38-s − 2.42e3·40-s + 2.17e3·41-s + ⋯ |
L(s) = 1 | + 0.152·2-s − 0.976·4-s + 0.795·5-s − 0.283·7-s − 0.301·8-s + 0.121·10-s + 0.753·11-s − 0.612·13-s − 0.0432·14-s + 0.930·16-s − 0.235·17-s + 0.871·19-s − 0.777·20-s + 0.114·22-s − 0.733·23-s − 0.367·25-s − 0.0935·26-s + 0.276·28-s + 0.185·29-s + 0.275·31-s + 0.443·32-s − 0.0359·34-s − 0.225·35-s − 1.40·37-s + 0.133·38-s − 0.240·40-s + 0.202·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 - 841T \) |
good | 2 | \( 1 - 0.863T + 32T^{2} \) |
| 5 | \( 1 - 44.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 36.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 302.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 373.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 280.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.86e3T + 6.43e6T^{2} \) |
| 31 | \( 1 - 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.17e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.67e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.56e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.49e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.35837666582656616323152764906, −9.601340756766831303698983194237, −8.994262418459754212276617627480, −7.76752041278643655851951938036, −6.43948943277212666908652155645, −5.48413935203021912384797176915, −4.42939799409509677001591378310, −3.19334675765172291648454955224, −1.56674063054787884816770728071, 0,
1.56674063054787884816770728071, 3.19334675765172291648454955224, 4.42939799409509677001591378310, 5.48413935203021912384797176915, 6.43948943277212666908652155645, 7.76752041278643655851951938036, 8.994262418459754212276617627480, 9.601340756766831303698983194237, 10.35837666582656616323152764906