| L(s) = 1 | − 1.16·2-s + 1.36·3-s − 0.649·4-s + 5-s − 1.58·6-s + 3.38·7-s + 3.07·8-s − 1.14·9-s − 1.16·10-s + 11-s − 0.884·12-s + 1.07·13-s − 3.94·14-s + 1.36·15-s − 2.28·16-s − 2.16·17-s + 1.33·18-s − 5.65·19-s − 0.649·20-s + 4.61·21-s − 1.16·22-s − 3.98·23-s + 4.19·24-s + 25-s − 1.25·26-s − 5.64·27-s − 2.20·28-s + ⋯ |
| L(s) = 1 | − 0.821·2-s + 0.786·3-s − 0.324·4-s + 0.447·5-s − 0.646·6-s + 1.28·7-s + 1.08·8-s − 0.381·9-s − 0.367·10-s + 0.301·11-s − 0.255·12-s + 0.299·13-s − 1.05·14-s + 0.351·15-s − 0.570·16-s − 0.526·17-s + 0.313·18-s − 1.29·19-s − 0.145·20-s + 1.00·21-s − 0.247·22-s − 0.830·23-s + 0.856·24-s + 0.200·25-s − 0.246·26-s − 1.08·27-s − 0.415·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 3 | \( 1 - 1.36T + 3T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 + 2.16T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 3.98T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 0.867T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 + 8.89T + 61T^{2} \) |
| 67 | \( 1 + 9.97T + 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 79 | \( 1 - 9.63T + 79T^{2} \) |
| 83 | \( 1 + 5.62T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 + 3.15T + 97T^{2} \) |
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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
−8.295549648932983534518082979395, −7.75491012300224103668453356575, −6.84007703047345687971543537058, −5.86684872610738024970867844683, −4.96342491465401433982575181910, −4.28955445682897265052432082911, −3.37557039765379424960541874622, −1.96000962369789925905134169292, −1.70883755119179168854314003677, 0,
1.70883755119179168854314003677, 1.96000962369789925905134169292, 3.37557039765379424960541874622, 4.28955445682897265052432082911, 4.96342491465401433982575181910, 5.86684872610738024970867844683, 6.84007703047345687971543537058, 7.75491012300224103668453356575, 8.295549648932983534518082979395
