| L(s) = 1 | + 2-s + 0.100·3-s + 4-s − 2.25·5-s + 0.100·6-s + 1.56·7-s + 8-s − 2.98·9-s − 2.25·10-s + 3.22·11-s + 0.100·12-s − 0.972·13-s + 1.56·14-s − 0.226·15-s + 16-s − 2.87·17-s − 2.98·18-s + 1.82·19-s − 2.25·20-s + 0.156·21-s + 3.22·22-s + 23-s + 0.100·24-s + 0.101·25-s − 0.972·26-s − 0.601·27-s + 1.56·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0579·3-s + 0.5·4-s − 1.01·5-s + 0.0409·6-s + 0.589·7-s + 0.353·8-s − 0.996·9-s − 0.714·10-s + 0.971·11-s + 0.0289·12-s − 0.269·13-s + 0.417·14-s − 0.0585·15-s + 0.250·16-s − 0.698·17-s − 0.704·18-s + 0.417·19-s − 0.505·20-s + 0.0341·21-s + 0.686·22-s + 0.208·23-s + 0.0204·24-s + 0.0203·25-s − 0.190·26-s − 0.115·27-s + 0.294·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 0.100T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 + 0.972T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 29 | \( 1 + 7.81T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 + 8.45T + 61T^{2} \) |
| 67 | \( 1 + 9.02T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 4.79T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
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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
−7.55345100271091269775257855132, −7.16736960114267326001137037340, −6.12699121379310816347708264865, −5.59831412015649264451915376042, −4.64944730713796225314002678292, −4.09245036411404396815871649935, −3.39192059048389117765118913804, −2.52139813080910978401848904692, −1.45922782604309696756919832189, 0,
1.45922782604309696756919832189, 2.52139813080910978401848904692, 3.39192059048389117765118913804, 4.09245036411404396815871649935, 4.64944730713796225314002678292, 5.59831412015649264451915376042, 6.12699121379310816347708264865, 7.16736960114267326001137037340, 7.55345100271091269775257855132
