| L(s) = 1 | + 2.25·3-s + 2.34·5-s − 2.23·7-s + 2.09·9-s − 4.83·11-s + 0.834·13-s + 5.28·15-s − 5.54·17-s + 1.83·19-s − 5.04·21-s + 7.33·23-s + 0.483·25-s − 2.05·27-s − 3.35·29-s − 0.777·31-s − 10.9·33-s − 5.23·35-s − 7.78·37-s + 1.88·39-s − 9.17·41-s + 11.5·43-s + 4.89·45-s + 1.92·47-s − 1.99·49-s − 12.5·51-s − 2.04·53-s − 11.3·55-s + ⋯ |
| L(s) = 1 | + 1.30·3-s + 1.04·5-s − 0.845·7-s + 0.696·9-s − 1.45·11-s + 0.231·13-s + 1.36·15-s − 1.34·17-s + 0.420·19-s − 1.10·21-s + 1.52·23-s + 0.0967·25-s − 0.395·27-s − 0.623·29-s − 0.139·31-s − 1.89·33-s − 0.885·35-s − 1.28·37-s + 0.301·39-s − 1.43·41-s + 1.76·43-s + 0.729·45-s + 0.280·47-s − 0.285·49-s − 1.75·51-s − 0.281·53-s − 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 - 0.834T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 + 0.777T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 2.04T + 53T^{2} \) |
| 59 | \( 1 + 0.668T + 59T^{2} \) |
| 61 | \( 1 + 8.64T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 3.57T + 71T^{2} \) |
| 73 | \( 1 - 5.69T + 73T^{2} \) |
| 79 | \( 1 - 8.38T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
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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
−7.42895578203820308081563590936, −6.98473310983976313363314202308, −6.07979225177183923086764393351, −5.44125463946272981404901833262, −4.68388209720231000445452650795, −3.58493720792756095507911954997, −2.93430560824856760204652702376, −2.42335277421438835180683341560, −1.63770770193908761206466838298, 0,
1.63770770193908761206466838298, 2.42335277421438835180683341560, 2.93430560824856760204652702376, 3.58493720792756095507911954997, 4.68388209720231000445452650795, 5.44125463946272981404901833262, 6.07979225177183923086764393351, 6.98473310983976313363314202308, 7.42895578203820308081563590936
