| L(s) = 1 | + 0.193·3-s + 1.98·5-s + 0.152·7-s − 2.96·9-s + 4.68·11-s + 3.34·13-s + 0.385·15-s + 1.27·17-s − 6.56·19-s + 0.0294·21-s + 4.53·23-s − 1.04·25-s − 1.15·27-s + 5.69·29-s + 6.91·31-s + 0.908·33-s + 0.302·35-s − 3.32·37-s + 0.648·39-s + 4.64·41-s + 3.48·43-s − 5.89·45-s + 0.481·47-s − 6.97·49-s + 0.247·51-s − 6.00·53-s + 9.32·55-s + ⋯ |
| L(s) = 1 | + 0.111·3-s + 0.889·5-s + 0.0575·7-s − 0.987·9-s + 1.41·11-s + 0.928·13-s + 0.0995·15-s + 0.309·17-s − 1.50·19-s + 0.00643·21-s + 0.946·23-s − 0.208·25-s − 0.222·27-s + 1.05·29-s + 1.24·31-s + 0.158·33-s + 0.0511·35-s − 0.547·37-s + 0.103·39-s + 0.726·41-s + 0.531·43-s − 0.878·45-s + 0.0702·47-s − 0.996·49-s + 0.0346·51-s − 0.824·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.846850679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.846850679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 0.193T + 3T^{2} \) |
| 5 | \( 1 - 1.98T + 5T^{2} \) |
| 7 | \( 1 - 0.152T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 - 6.91T + 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 - 0.481T + 47T^{2} \) |
| 53 | \( 1 + 6.00T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 + 0.426T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 - 4.54T + 79T^{2} \) |
| 83 | \( 1 + 4.67T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 + 2.96T + 97T^{2} \) |
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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
−8.072034393890848715637628728788, −6.89548010060766278652147043337, −6.22032604136647996647898750356, −6.09690942672395794456796691046, −5.06604070148410280913544811071, −4.27195366853275758292203050781, −3.46414883377330126802570621428, −2.64181507455567811498378086095, −1.76414135864013114799390701731, −0.866508348561201105709479599774,
0.866508348561201105709479599774, 1.76414135864013114799390701731, 2.64181507455567811498378086095, 3.46414883377330126802570621428, 4.27195366853275758292203050781, 5.06604070148410280913544811071, 6.09690942672395794456796691046, 6.22032604136647996647898750356, 6.89548010060766278652147043337, 8.072034393890848715637628728788
