| L(s) = 1 | + 2.56·3-s + 5-s + 2.56·7-s + 3.56·9-s − 4·11-s + 5.68·13-s + 2.56·15-s + 3.43·17-s + 19-s + 6.56·21-s − 7.68·23-s + 25-s + 1.43·27-s − 5.68·29-s + 5.12·31-s − 10.2·33-s + 2.56·35-s − 6·37-s + 14.5·39-s + 12.2·41-s + 2.87·43-s + 3.56·45-s − 6.24·47-s − 0.438·49-s + 8.80·51-s − 4.56·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s + 0.968·7-s + 1.18·9-s − 1.20·11-s + 1.57·13-s + 0.661·15-s + 0.833·17-s + 0.229·19-s + 1.43·21-s − 1.60·23-s + 0.200·25-s + 0.276·27-s − 1.05·29-s + 0.920·31-s − 1.78·33-s + 0.432·35-s − 0.986·37-s + 2.33·39-s + 1.91·41-s + 0.438·43-s + 0.530·45-s − 0.911·47-s − 0.0626·49-s + 1.23·51-s − 0.626·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.407202442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.407202442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.68T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−9.363333831392560549808861387661, −8.456911825817670259377409201241, −8.045589858216953366856602737288, −7.50961840951425900209996324917, −6.11025723091500428254980998255, −5.36323370694267412419401940282, −4.18196536370930174701442462772, −3.33513865792707630051798242620, −2.34125634136615220181658296454, −1.46752139324496923339717940844,
1.46752139324496923339717940844, 2.34125634136615220181658296454, 3.33513865792707630051798242620, 4.18196536370930174701442462772, 5.36323370694267412419401940282, 6.11025723091500428254980998255, 7.50961840951425900209996324917, 8.045589858216953366856602737288, 8.456911825817670259377409201241, 9.363333831392560549808861387661
