| L(s) = 1 | + 2.23·3-s + 3.29·5-s + 2.75·7-s + 2.01·9-s + 1.05·11-s + 1.68·13-s + 7.37·15-s + 5.50·19-s + 6.17·21-s + 0.493·23-s + 5.83·25-s − 2.20·27-s + 9.81·29-s + 2.16·31-s + 2.37·33-s + 9.08·35-s + 3.43·37-s + 3.76·39-s − 7.08·41-s + 10.1·43-s + 6.63·45-s − 9.02·47-s + 0.612·49-s − 13.9·53-s + 3.48·55-s + 12.3·57-s − 11.2·59-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 1.47·5-s + 1.04·7-s + 0.671·9-s + 0.319·11-s + 0.466·13-s + 1.90·15-s + 1.26·19-s + 1.34·21-s + 0.102·23-s + 1.16·25-s − 0.424·27-s + 1.82·29-s + 0.388·31-s + 0.413·33-s + 1.53·35-s + 0.565·37-s + 0.603·39-s − 1.10·41-s + 1.54·43-s + 0.988·45-s − 1.31·47-s + 0.0875·49-s − 1.91·53-s + 0.470·55-s + 1.63·57-s − 1.45·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.981908068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.981908068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 19 | \( 1 - 5.50T + 19T^{2} \) |
| 23 | \( 1 - 0.493T + 23T^{2} \) |
| 29 | \( 1 - 9.81T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 + 7.08T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 9.02T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 + 0.0130T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 8.48T + 97T^{2} \) |
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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
−7.961842235388927299779024349558, −7.12064617279749930946933084286, −6.32658604699049510412166527312, −5.68133242239036652973100741494, −4.91282138296996375100388499475, −4.24814112790282774746762082970, −3.05115171027037012023212058808, −2.76010118707205810579957807953, −1.63344444490512716842472439845, −1.34403903604177734799707126512,
1.34403903604177734799707126512, 1.63344444490512716842472439845, 2.76010118707205810579957807953, 3.05115171027037012023212058808, 4.24814112790282774746762082970, 4.91282138296996375100388499475, 5.68133242239036652973100741494, 6.32658604699049510412166527312, 7.12064617279749930946933084286, 7.961842235388927299779024349558
