| L(s) = 1 | − 0.818·2-s − 0.529·3-s − 0.330·4-s − 1.94·5-s + 0.433·6-s + 0.676·7-s + 1.08·8-s − 0.719·9-s + 1.59·10-s − 1.85·11-s + 0.175·12-s − 0.553·14-s + 1.03·15-s − 0.560·16-s + 0.588·18-s + 0.643·20-s − 0.358·21-s + 1.51·22-s − 0.576·24-s + 2.79·25-s + 0.911·27-s − 0.223·28-s + 0.380·29-s − 0.844·30-s − 0.630·32-s + 0.983·33-s − 1.31·35-s + ⋯ |
| L(s) = 1 | − 0.818·2-s − 0.529·3-s − 0.330·4-s − 1.94·5-s + 0.433·6-s + 0.676·7-s + 1.08·8-s − 0.719·9-s + 1.59·10-s − 1.85·11-s + 0.175·12-s − 0.553·14-s + 1.03·15-s − 0.560·16-s + 0.588·18-s + 0.643·20-s − 0.358·21-s + 1.51·22-s − 0.576·24-s + 2.79·25-s + 0.911·27-s − 0.223·28-s + 0.380·29-s − 0.844·30-s − 0.630·32-s + 0.983·33-s − 1.31·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2333905852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2333905852\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1151 | \( 1 - T \) |
| good | 2 | \( 1 + 0.818T + T^{2} \) |
| 3 | \( 1 + 0.529T + T^{2} \) |
| 5 | \( 1 + 1.94T + T^{2} \) |
| 7 | \( 1 - 0.676T + T^{2} \) |
| 11 | \( 1 + 1.85T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.380T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.08T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.90T + T^{2} \) |
| 47 | \( 1 - 1.21T + T^{2} \) |
| 53 | \( 1 - 1.97T + T^{2} \) |
| 59 | \( 1 - 1.79T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.99T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.54T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.28194803039850276473525931906, −8.676849473023188598624043102854, −8.505969046340167790368895207109, −7.61580434928780143614893213804, −7.26309800379360729647696423694, −5.53223827204979166462054637231, −4.84252493089989710757242026970, −4.02102887455522017563677366563, −2.72551192401937624167456113134, −0.60545956681672527135147983089,
0.60545956681672527135147983089, 2.72551192401937624167456113134, 4.02102887455522017563677366563, 4.84252493089989710757242026970, 5.53223827204979166462054637231, 7.26309800379360729647696423694, 7.61580434928780143614893213804, 8.505969046340167790368895207109, 8.676849473023188598624043102854, 10.28194803039850276473525931906
