| L(s) = 1 | + 2-s + 3-s + 4-s − 0.867·5-s + 6-s − 7-s + 8-s + 9-s − 0.867·10-s + 5.35·11-s + 12-s − 14-s − 0.867·15-s + 16-s + 4.35·17-s + 18-s − 0.607·19-s − 0.867·20-s − 21-s + 5.35·22-s + 0.637·23-s + 24-s − 4.24·25-s + 27-s − 28-s − 2.55·29-s − 0.867·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.388·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.274·10-s + 1.61·11-s + 0.288·12-s − 0.267·14-s − 0.224·15-s + 0.250·16-s + 1.05·17-s + 0.235·18-s − 0.139·19-s − 0.194·20-s − 0.218·21-s + 1.14·22-s + 0.132·23-s + 0.204·24-s − 0.849·25-s + 0.192·27-s − 0.188·28-s − 0.473·29-s − 0.158·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.319314797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.319314797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.867T + 5T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 + 0.607T + 19T^{2} \) |
| 23 | \( 1 - 0.637T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 + 9.33T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 5.67T + 73T^{2} \) |
| 79 | \( 1 - 7.81T + 79T^{2} \) |
| 83 | \( 1 - 5.90T + 83T^{2} \) |
| 89 | \( 1 - 0.515T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.984922349864434688847281275376, −7.06554967781233111402850509682, −6.58408713592246773122888819385, −5.89593576223674457075792076113, −4.98621287604800269756115770512, −4.10229661428884574278390849712, −3.66743422576824851457417048142, −2.98503447213174522029603590702, −1.93127574320986776694076065244, −0.969899334244049287901954485480,
0.969899334244049287901954485480, 1.93127574320986776694076065244, 2.98503447213174522029603590702, 3.66743422576824851457417048142, 4.10229661428884574278390849712, 4.98621287604800269756115770512, 5.89593576223674457075792076113, 6.58408713592246773122888819385, 7.06554967781233111402850509682, 7.984922349864434688847281275376
