L(s) = 1 | − 2-s + 0.405·3-s + 4-s − 2.02·5-s − 0.405·6-s − 4.23·7-s − 8-s − 2.83·9-s + 2.02·10-s + 11-s + 0.405·12-s + 3.21·13-s + 4.23·14-s − 0.819·15-s + 16-s − 2.00·17-s + 2.83·18-s − 2.02·20-s − 1.71·21-s − 22-s − 2.37·23-s − 0.405·24-s − 0.906·25-s − 3.21·26-s − 2.36·27-s − 4.23·28-s + 2.15·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.233·3-s + 0.5·4-s − 0.904·5-s − 0.165·6-s − 1.60·7-s − 0.353·8-s − 0.945·9-s + 0.639·10-s + 0.301·11-s + 0.116·12-s + 0.892·13-s + 1.13·14-s − 0.211·15-s + 0.250·16-s − 0.485·17-s + 0.668·18-s − 0.452·20-s − 0.374·21-s − 0.213·22-s − 0.494·23-s − 0.0827·24-s − 0.181·25-s − 0.631·26-s − 0.455·27-s − 0.800·28-s + 0.400·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.405T + 3T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 13 | \( 1 - 3.21T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 2.15T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 2.99T + 61T^{2} \) |
| 67 | \( 1 - 1.86T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 + 5.29T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 + 6.45T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−7.77742661410757894873894057024, −6.73614418346268713427285976603, −6.27486302094056769900781606887, −5.79101111831183977628952393649, −4.41052891024045094835887814889, −3.71978049235899281861621199381, −3.05711745461597912381811617581, −2.40457716172714322768270611456, −0.902421650037717782136201899671, 0,
0.902421650037717782136201899671, 2.40457716172714322768270611456, 3.05711745461597912381811617581, 3.71978049235899281861621199381, 4.41052891024045094835887814889, 5.79101111831183977628952393649, 6.27486302094056769900781606887, 6.73614418346268713427285976603, 7.77742661410757894873894057024