| L(s) = 1 | − 2.73·2-s + 3-s + 5.50·4-s + 2.84·5-s − 2.73·6-s − 3.93·7-s − 9.58·8-s + 9-s − 7.78·10-s − 11-s + 5.50·12-s + 10.7·14-s + 2.84·15-s + 15.2·16-s + 3.81·17-s − 2.73·18-s + 2.94·19-s + 15.6·20-s − 3.93·21-s + 2.73·22-s − 1.89·23-s − 9.58·24-s + 3.07·25-s + 27-s − 21.6·28-s − 2.09·29-s − 7.78·30-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.75·4-s + 1.27·5-s − 1.11·6-s − 1.48·7-s − 3.38·8-s + 0.333·9-s − 2.46·10-s − 0.301·11-s + 1.58·12-s + 2.87·14-s + 0.733·15-s + 3.81·16-s + 0.926·17-s − 0.645·18-s + 0.674·19-s + 3.49·20-s − 0.857·21-s + 0.583·22-s − 0.395·23-s − 1.95·24-s + 0.614·25-s + 0.192·27-s − 4.08·28-s − 0.389·29-s − 1.42·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 2.84T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 41 | \( 1 + 6.35T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 1.57T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.044545333471564025909506275058, −7.15015501634060252107063393008, −6.64632158672489329068713394782, −6.03739762323522552777652384289, −5.29844280461201546924575155875, −3.36994325057554322143589480833, −3.00827588802802444326164293418, −2.05733419985698124414175194443, −1.32014839304408894187199495511, 0,
1.32014839304408894187199495511, 2.05733419985698124414175194443, 3.00827588802802444326164293418, 3.36994325057554322143589480833, 5.29844280461201546924575155875, 6.03739762323522552777652384289, 6.64632158672489329068713394782, 7.15015501634060252107063393008, 8.044545333471564025909506275058
