| L(s) = 1 | − 1.04·2-s + 0.868·3-s − 0.902·4-s − 5-s − 0.909·6-s + 0.626·7-s + 3.04·8-s − 2.24·9-s + 1.04·10-s + 0.205·11-s − 0.783·12-s + 0.108·13-s − 0.656·14-s − 0.868·15-s − 1.38·16-s + 0.696·17-s + 2.35·18-s + 0.0951·19-s + 0.902·20-s + 0.544·21-s − 0.214·22-s + 2.64·24-s + 25-s − 0.113·26-s − 4.55·27-s − 0.565·28-s − 1.96·29-s + ⋯ |
| L(s) = 1 | − 0.740·2-s + 0.501·3-s − 0.451·4-s − 0.447·5-s − 0.371·6-s + 0.236·7-s + 1.07·8-s − 0.748·9-s + 0.331·10-s + 0.0618·11-s − 0.226·12-s + 0.0300·13-s − 0.175·14-s − 0.224·15-s − 0.345·16-s + 0.169·17-s + 0.554·18-s + 0.0218·19-s + 0.201·20-s + 0.118·21-s − 0.0457·22-s + 0.539·24-s + 0.200·25-s − 0.0222·26-s − 0.876·27-s − 0.106·28-s − 0.364·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 3 | \( 1 - 0.868T + 3T^{2} \) |
| 7 | \( 1 - 0.626T + 7T^{2} \) |
| 11 | \( 1 - 0.205T + 11T^{2} \) |
| 13 | \( 1 - 0.108T + 13T^{2} \) |
| 17 | \( 1 - 0.696T + 17T^{2} \) |
| 19 | \( 1 - 0.0951T + 19T^{2} \) |
| 29 | \( 1 + 1.96T + 29T^{2} \) |
| 31 | \( 1 - 7.26T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 + 5.79T + 53T^{2} \) |
| 59 | \( 1 + 6.02T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 6.45T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 2.96T + 97T^{2} \) |
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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
−8.530663369717295544624907820973, −7.906404290034679681464270735955, −7.37995638587418655325249600837, −6.25001715174663959890752643065, −5.27704078166292854861693265500, −4.44309577732141104047523411285, −3.59572192019620748942555794098, −2.59908306882974592218331905709, −1.32876011457603924158945754696, 0,
1.32876011457603924158945754696, 2.59908306882974592218331905709, 3.59572192019620748942555794098, 4.44309577732141104047523411285, 5.27704078166292854861693265500, 6.25001715174663959890752643065, 7.37995638587418655325249600837, 7.906404290034679681464270735955, 8.530663369717295544624907820973
