L(s) = 1 | − 2-s − 3.27·3-s + 4-s − 0.103·5-s + 3.27·6-s + 0.553·7-s − 8-s + 7.69·9-s + 0.103·10-s + 0.813·11-s − 3.27·12-s + 6.58·13-s − 0.553·14-s + 0.339·15-s + 16-s + 5.83·17-s − 7.69·18-s + 3.59·19-s − 0.103·20-s − 1.80·21-s − 0.813·22-s − 23-s + 3.27·24-s − 4.98·25-s − 6.58·26-s − 15.3·27-s + 0.553·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.88·3-s + 0.5·4-s − 0.0464·5-s + 1.33·6-s + 0.209·7-s − 0.353·8-s + 2.56·9-s + 0.0328·10-s + 0.245·11-s − 0.944·12-s + 1.82·13-s − 0.147·14-s + 0.0876·15-s + 0.250·16-s + 1.41·17-s − 1.81·18-s + 0.825·19-s − 0.0232·20-s − 0.394·21-s − 0.173·22-s − 0.208·23-s + 0.667·24-s − 0.997·25-s − 1.29·26-s − 2.95·27-s + 0.104·28-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 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 + 0.103T + 5T^{2} \) |
| 7 | \( 1 - 0.553T + 7T^{2} \) |
| 11 | \( 1 - 0.813T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 7.85T + 31T^{2} \) |
| 37 | \( 1 - 0.849T + 37T^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 + 6.56T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 + 3.93T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 0.873T + 67T^{2} \) |
| 71 | \( 1 - 6.26T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 5.08T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 + 5.53T + 97T^{2} \) |
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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
−7.63501998189065409227871433363, −6.97196376782227847286440504862, −6.19024401006913602771079921763, −5.68570587523050565732048130055, −5.24538558437079832441686360071, −4.02063789011396172232568991157, −3.46367226733141550543558406357, −1.60868488964204811591381092580, −1.20713432643807350331306966009, 0,
1.20713432643807350331306966009, 1.60868488964204811591381092580, 3.46367226733141550543558406357, 4.02063789011396172232568991157, 5.24538558437079832441686360071, 5.68570587523050565732048130055, 6.19024401006913602771079921763, 6.97196376782227847286440504862, 7.63501998189065409227871433363