| L(s) = 1 | − 3-s + 5-s + 1.17·7-s + 9-s − 2·11-s + 0.0917·13-s − 15-s − 6.04·17-s − 1.17·21-s + 1.70·23-s + 25-s − 27-s + 7.95·29-s − 31-s + 2·33-s + 1.17·35-s − 3.35·37-s − 0.0917·39-s − 4.68·41-s + 0.738·43-s + 45-s + 9.12·47-s − 5.63·49-s + 6.04·51-s + 4.97·53-s − 2·55-s − 11.0·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.442·7-s + 0.333·9-s − 0.603·11-s + 0.0254·13-s − 0.258·15-s − 1.46·17-s − 0.255·21-s + 0.356·23-s + 0.200·25-s − 0.192·27-s + 1.47·29-s − 0.179·31-s + 0.348·33-s + 0.197·35-s − 0.551·37-s − 0.0146·39-s − 0.730·41-s + 0.112·43-s + 0.149·45-s + 1.33·47-s − 0.804·49-s + 0.847·51-s + 0.682·53-s − 0.269·55-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.0917T + 13T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 - 7.95T + 29T^{2} \) |
| 37 | \( 1 + 3.35T + 37T^{2} \) |
| 41 | \( 1 + 4.68T + 41T^{2} \) |
| 43 | \( 1 - 0.738T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−7.41963951397145621851859305053, −6.78060700561801050739994293033, −6.18429280038503082016875287505, −5.38661539576505188699813364066, −4.78954391020416043053887302641, −4.19216974095262438641336417606, −2.99482942769004195950517569942, −2.20896888689304046289931564222, −1.26933595284785679364579699419, 0,
1.26933595284785679364579699419, 2.20896888689304046289931564222, 2.99482942769004195950517569942, 4.19216974095262438641336417606, 4.78954391020416043053887302641, 5.38661539576505188699813364066, 6.18429280038503082016875287505, 6.78060700561801050739994293033, 7.41963951397145621851859305053
