| L(s) = 1 | − 2.72·2-s − 1.81·3-s + 5.41·4-s + 4.95·6-s − 1.94·7-s − 9.30·8-s + 0.307·9-s + 3.95·11-s − 9.85·12-s + 4.17·13-s + 5.29·14-s + 14.5·16-s + 7.42·17-s − 0.838·18-s + 6.85·19-s + 3.53·21-s − 10.7·22-s − 4.18·23-s + 16.9·24-s − 11.3·26-s + 4.89·27-s − 10.5·28-s − 7.88·29-s − 1.11·31-s − 20.8·32-s − 7.19·33-s − 20.2·34-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 1.05·3-s + 2.70·4-s + 2.02·6-s − 0.734·7-s − 3.28·8-s + 0.102·9-s + 1.19·11-s − 2.84·12-s + 1.15·13-s + 1.41·14-s + 3.62·16-s + 1.80·17-s − 0.197·18-s + 1.57·19-s + 0.771·21-s − 2.29·22-s − 0.872·23-s + 3.45·24-s − 2.23·26-s + 0.942·27-s − 1.98·28-s − 1.46·29-s − 0.201·31-s − 3.69·32-s − 1.25·33-s − 3.46·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 - 7.42T + 17T^{2} \) |
| 19 | \( 1 - 6.85T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 + 7.88T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 - 0.972T + 37T^{2} \) |
| 41 | \( 1 + 7.49T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 5.11T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 - 2.08T + 67T^{2} \) |
| 71 | \( 1 + 6.94T + 71T^{2} \) |
| 73 | \( 1 - 9.55T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.892701023864232826559067617172, −7.00205638090267848250132318099, −6.51917626992937313023390624636, −5.86893813760145263111253305448, −5.38853052626592164211071840942, −3.55245423921824812320240602622, −3.22620322546013340745451161829, −1.61400990831628449434603439579, −1.08626307382773142582439468003, 0,
1.08626307382773142582439468003, 1.61400990831628449434603439579, 3.22620322546013340745451161829, 3.55245423921824812320240602622, 5.38853052626592164211071840942, 5.86893813760145263111253305448, 6.51917626992937313023390624636, 7.00205638090267848250132318099, 7.892701023864232826559067617172
