| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s − 30-s − 31-s − 32-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 29-s − 30-s − 31-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2219 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2219 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.240811471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.240811471\) |
| \(L(1)\) |
\(\approx\) |
\(1.600598712\) |
| \(L(1)\) |
\(\approx\) |
\(1.600598712\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 317 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.498687693734588679893586248480, −18.61454793566656098033374454633, −18.38595077896106736999718159406, −17.45655703387417440318035871315, −16.664653144521369810654688315881, −16.16628617180335792614241292302, −15.136541007033498411689669918331, −14.52673065726978867056964566612, −13.89673443239879049702466142771, −13.032543363606489641736571932505, −12.279547919169806724395549328085, −11.20131017016852588268474413188, −10.556157517066609034506824766715, −9.5427119666342741664892369187, −9.29238476874812086511471640030, −8.699965201573715626741391203971, −7.63457882147232823150581657742, −7.1380807942850836689558146405, −6.13689266182158702579758688818, −5.50571643124434323538568704310, −4.00172007553361077719462787462, −3.1912422308395478175167562351, −2.434618813611404005868177228271, −1.30400146794627423483069024097, −1.1406550413878926907090100700,
1.1406550413878926907090100700, 1.30400146794627423483069024097, 2.434618813611404005868177228271, 3.1912422308395478175167562351, 4.00172007553361077719462787462, 5.50571643124434323538568704310, 6.13689266182158702579758688818, 7.1380807942850836689558146405, 7.63457882147232823150581657742, 8.699965201573715626741391203971, 9.29238476874812086511471640030, 9.5427119666342741664892369187, 10.556157517066609034506824766715, 11.20131017016852588268474413188, 12.279547919169806724395549328085, 13.032543363606489641736571932505, 13.89673443239879049702466142771, 14.52673065726978867056964566612, 15.136541007033498411689669918331, 16.16628617180335792614241292302, 16.664653144521369810654688315881, 17.45655703387417440318035871315, 18.38595077896106736999718159406, 18.61454793566656098033374454633, 19.498687693734588679893586248480
