| L(s) = 1 | − 1.95·3-s + 0.337·5-s − 0.830·7-s + 0.835·9-s + 5.82·11-s + 4.14·13-s − 0.660·15-s − 4.44·17-s − 1.68·19-s + 1.62·21-s + 6.61·23-s − 4.88·25-s + 4.23·27-s − 6.73·29-s − 0.271·31-s − 11.4·33-s − 0.280·35-s − 4.23·37-s − 8.12·39-s − 8.53·41-s + 0.824·43-s + 0.281·45-s − 8.45·47-s − 6.30·49-s + 8.70·51-s − 2.66·53-s + 1.96·55-s + ⋯ |
| L(s) = 1 | − 1.13·3-s + 0.150·5-s − 0.313·7-s + 0.278·9-s + 1.75·11-s + 1.15·13-s − 0.170·15-s − 1.07·17-s − 0.387·19-s + 0.355·21-s + 1.38·23-s − 0.977·25-s + 0.815·27-s − 1.25·29-s − 0.0487·31-s − 1.98·33-s − 0.0473·35-s − 0.696·37-s − 1.30·39-s − 1.33·41-s + 0.125·43-s + 0.0419·45-s − 1.23·47-s − 0.901·49-s + 1.21·51-s − 0.365·53-s + 0.265·55-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 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 0.337T + 5T^{2} \) |
| 7 | \( 1 + 0.830T + 7T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 0.271T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 - 0.824T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.05443858475267100155818181159, −6.66474754860504983156802490671, −6.20810359891277084084018717315, −5.51654189534365290622851974849, −4.75989733678136319596973884554, −3.89594598138993985196785944962, −3.36461664780863573364165340169, −1.95642462309303095627666765915, −1.16508087988359572997944701976, 0,
1.16508087988359572997944701976, 1.95642462309303095627666765915, 3.36461664780863573364165340169, 3.89594598138993985196785944962, 4.75989733678136319596973884554, 5.51654189534365290622851974849, 6.20810359891277084084018717315, 6.66474754860504983156802490671, 7.05443858475267100155818181159
