| L(s) = 1 | − 2-s + 3.22·3-s + 4-s − 3.48·5-s − 3.22·6-s + 7-s − 8-s + 7.39·9-s + 3.48·10-s − 4.14·11-s + 3.22·12-s + 3.72·13-s − 14-s − 11.2·15-s + 16-s + 3.53·17-s − 7.39·18-s − 6.23·19-s − 3.48·20-s + 3.22·21-s + 4.14·22-s − 1.88·23-s − 3.22·24-s + 7.17·25-s − 3.72·26-s + 14.1·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s − 1.56·5-s − 1.31·6-s + 0.377·7-s − 0.353·8-s + 2.46·9-s + 1.10·10-s − 1.24·11-s + 0.930·12-s + 1.03·13-s − 0.267·14-s − 2.90·15-s + 0.250·16-s + 0.857·17-s − 1.74·18-s − 1.43·19-s − 0.780·20-s + 0.703·21-s + 0.883·22-s − 0.392·23-s − 0.658·24-s + 1.43·25-s − 0.730·26-s + 2.72·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 1.88T + 23T^{2} \) |
| 29 | \( 1 + 9.33T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + 0.104T + 53T^{2} \) |
| 59 | \( 1 + 8.27T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 8.34T + 67T^{2} \) |
| 71 | \( 1 - 6.81T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−8.029089047965004145693512027519, −7.46006039196228186278405265942, −6.84439773878273116877926610747, −5.56682007951907963155220289607, −4.42459120307803237718947670141, −3.72528181824845204875089322078, −3.29160772389669698478897812593, −2.32158158315875161254349765040, −1.50457720779101488876073897150, 0,
1.50457720779101488876073897150, 2.32158158315875161254349765040, 3.29160772389669698478897812593, 3.72528181824845204875089322078, 4.42459120307803237718947670141, 5.56682007951907963155220289607, 6.84439773878273116877926610747, 7.46006039196228186278405265942, 8.029089047965004145693512027519
