| L(s) = 1 | + 2-s + 0.780·3-s + 4-s − 0.582·5-s + 0.780·6-s + 2.23·7-s + 8-s − 2.39·9-s − 0.582·10-s − 2.03·11-s + 0.780·12-s − 1.44·13-s + 2.23·14-s − 0.455·15-s + 16-s − 6.79·17-s − 2.39·18-s + 5.55·19-s − 0.582·20-s + 1.74·21-s − 2.03·22-s − 23-s + 0.780·24-s − 4.66·25-s − 1.44·26-s − 4.20·27-s + 2.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.450·3-s + 0.5·4-s − 0.260·5-s + 0.318·6-s + 0.845·7-s + 0.353·8-s − 0.796·9-s − 0.184·10-s − 0.614·11-s + 0.225·12-s − 0.400·13-s + 0.597·14-s − 0.117·15-s + 0.250·16-s − 1.64·17-s − 0.563·18-s + 1.27·19-s − 0.130·20-s + 0.381·21-s − 0.434·22-s − 0.208·23-s + 0.159·24-s − 0.932·25-s − 0.283·26-s − 0.810·27-s + 0.422·28-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 | \( 1 - 0.780T + 3T^{2} \) |
| 5 | \( 1 + 0.582T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 + 0.200T + 41T^{2} \) |
| 43 | \( 1 + 0.647T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 2.26T + 61T^{2} \) |
| 67 | \( 1 + 2.42T + 67T^{2} \) |
| 71 | \( 1 - 3.20T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 6.47T + 97T^{2} \) |
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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
−7.64225293132690815768618624551, −7.17803211647647158084420827311, −6.14609444667513416170022702964, −5.44503034834530226454478426035, −4.82083259859177574842677998470, −4.10513557439129875905551409880, −3.18675914705034779874028305111, −2.47837962472506755598786870884, −1.69010092867933572609015594399, 0,
1.69010092867933572609015594399, 2.47837962472506755598786870884, 3.18675914705034779874028305111, 4.10513557439129875905551409880, 4.82083259859177574842677998470, 5.44503034834530226454478426035, 6.14609444667513416170022702964, 7.17803211647647158084420827311, 7.64225293132690815768618624551
