| L(s) = 1 | + 2-s − 3-s + 4-s − 4.08·5-s − 6-s − 0.153·7-s + 8-s + 9-s − 4.08·10-s + 4.40·11-s − 12-s + 0.950·13-s − 0.153·14-s + 4.08·15-s + 16-s − 17-s + 18-s − 3.15·19-s − 4.08·20-s + 0.153·21-s + 4.40·22-s + 0.350·23-s − 24-s + 11.7·25-s + 0.950·26-s − 27-s − 0.153·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.82·5-s − 0.408·6-s − 0.0581·7-s + 0.353·8-s + 0.333·9-s − 1.29·10-s + 1.32·11-s − 0.288·12-s + 0.263·13-s − 0.0410·14-s + 1.05·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.724·19-s − 0.913·20-s + 0.0335·21-s + 0.938·22-s + 0.0731·23-s − 0.204·24-s + 2.34·25-s + 0.186·26-s − 0.192·27-s − 0.0290·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.604695117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604695117\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 + 4.08T + 5T^{2} \) |
| 7 | \( 1 + 0.153T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 - 0.950T + 13T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 - 0.350T + 23T^{2} \) |
| 29 | \( 1 - 5.57T + 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 - 0.0233T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 + 7.47T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 4.18T + 53T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 7.84T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.108628012859853990040380785967, −7.03453876323204093738280975521, −6.75839675311341651554360289392, −6.05564760452469211216940558430, −4.85171589407651522844450807701, −4.52250224727748721847143356488, −3.65747847144596022284117830664, −3.30922563552055077550420031661, −1.79993681441941106128517571576, −0.62130052108315092691545390764,
0.62130052108315092691545390764, 1.79993681441941106128517571576, 3.30922563552055077550420031661, 3.65747847144596022284117830664, 4.52250224727748721847143356488, 4.85171589407651522844450807701, 6.05564760452469211216940558430, 6.75839675311341651554360289392, 7.03453876323204093738280975521, 8.108628012859853990040380785967
