| L(s) = 1 | + 2-s − 1.20·3-s + 4-s − 1.11·5-s − 1.20·6-s − 0.129·7-s + 8-s − 1.53·9-s − 1.11·10-s + 3.11·11-s − 1.20·12-s + 6.27·13-s − 0.129·14-s + 1.35·15-s + 16-s − 1.75·17-s − 1.53·18-s − 5.63·19-s − 1.11·20-s + 0.157·21-s + 3.11·22-s + 23-s − 1.20·24-s − 3.75·25-s + 6.27·26-s + 5.48·27-s − 0.129·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.698·3-s + 0.5·4-s − 0.499·5-s − 0.493·6-s − 0.0491·7-s + 0.353·8-s − 0.511·9-s − 0.353·10-s + 0.939·11-s − 0.349·12-s + 1.73·13-s − 0.0347·14-s + 0.349·15-s + 0.250·16-s − 0.424·17-s − 0.362·18-s − 1.29·19-s − 0.249·20-s + 0.0343·21-s + 0.664·22-s + 0.208·23-s − 0.246·24-s − 0.750·25-s + 1.22·26-s + 1.05·27-s − 0.0245·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.922798307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.922798307\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.20T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 0.129T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 - 6.27T + 13T^{2} \) |
| 17 | \( 1 + 1.75T + 17T^{2} \) |
| 19 | \( 1 + 5.63T + 19T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 8.19T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 - 0.205T + 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 - 4.62T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 - 9.72T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.00107T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−9.719137727289159802945639764449, −8.566909429940494146506671969011, −8.142784572579072676879277000757, −6.62654226000434749080685989337, −6.37516458114304363544125644356, −5.55210930657825875590653075488, −4.31575951234922081929715694866, −3.86545359783861631042124314579, −2.56986250941755044193579772037, −0.980527162513218390717693819316,
0.980527162513218390717693819316, 2.56986250941755044193579772037, 3.86545359783861631042124314579, 4.31575951234922081929715694866, 5.55210930657825875590653075488, 6.37516458114304363544125644356, 6.62654226000434749080685989337, 8.142784572579072676879277000757, 8.566909429940494146506671969011, 9.719137727289159802945639764449
