| L(s) = 1 | − 0.585·3-s − 5-s + 4.82·7-s − 2.65·9-s − 2·11-s − 2.24·13-s + 0.585·15-s − 4.82·17-s − 19-s − 2.82·21-s + 6·23-s + 25-s + 3.31·27-s − 10.4·29-s + 1.17·31-s + 1.17·33-s − 4.82·35-s + 10.2·37-s + 1.31·39-s − 7.65·41-s − 0.828·43-s + 2.65·45-s − 0.828·47-s + 16.3·49-s + 2.82·51-s + 5.07·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.338·3-s − 0.447·5-s + 1.82·7-s − 0.885·9-s − 0.603·11-s − 0.621·13-s + 0.151·15-s − 1.17·17-s − 0.229·19-s − 0.617·21-s + 1.25·23-s + 0.200·25-s + 0.637·27-s − 1.94·29-s + 0.210·31-s + 0.203·33-s − 0.816·35-s + 1.68·37-s + 0.210·39-s − 1.19·41-s − 0.126·43-s + 0.396·45-s − 0.120·47-s + 2.33·49-s + 0.396·51-s + 0.696·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387196386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387196386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 2.34T + 61T^{2} \) |
| 67 | \( 1 + 0.585T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−7.947861460022356608331365400666, −7.56816008764371744985267929475, −6.73293544784854002754262875083, −5.76684419830009614012899265730, −4.98504766596448602259716081332, −4.77616891203080074289430267124, −3.75088571733376059673956504636, −2.59335884053265231702517697252, −1.93929606032299414233200589226, −0.61401219380394009344543808522,
0.61401219380394009344543808522, 1.93929606032299414233200589226, 2.59335884053265231702517697252, 3.75088571733376059673956504636, 4.77616891203080074289430267124, 4.98504766596448602259716081332, 5.76684419830009614012899265730, 6.73293544784854002754262875083, 7.56816008764371744985267929475, 7.947861460022356608331365400666
