| L(s) = 1 | − 3-s − 3.23·5-s − 4·7-s + 9-s + 6.47·11-s + 4.47·13-s + 3.23·15-s − 0.763·17-s + 4·21-s + 1.23·23-s + 5.47·25-s − 27-s + 9.70·29-s − 1.52·31-s − 6.47·33-s + 12.9·35-s − 37-s − 4.47·39-s − 10.9·41-s − 2.47·43-s − 3.23·45-s − 8·47-s + 9·49-s + 0.763·51-s − 8.47·53-s − 20.9·55-s − 10.1·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.44·5-s − 1.51·7-s + 0.333·9-s + 1.95·11-s + 1.24·13-s + 0.835·15-s − 0.185·17-s + 0.872·21-s + 0.257·23-s + 1.09·25-s − 0.192·27-s + 1.80·29-s − 0.274·31-s − 1.12·33-s + 2.18·35-s − 0.164·37-s − 0.716·39-s − 1.70·41-s − 0.376·43-s − 0.482·45-s − 1.16·47-s + 1.28·49-s + 0.106·51-s − 1.16·53-s − 2.82·55-s − 1.32·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9289147945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9289147945\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.987701642509487313713570245448, −6.87661178761912228051069824149, −6.55921701220043058458824166469, −6.24565089420628760623605522956, −4.97527985474830191533505902412, −4.17728175990688111935683440489, −3.54246407423164886663315029341, −3.22248312428592357133900169049, −1.46936854159902414952259544481, −0.54096091468856009548009605746,
0.54096091468856009548009605746, 1.46936854159902414952259544481, 3.22248312428592357133900169049, 3.54246407423164886663315029341, 4.17728175990688111935683440489, 4.97527985474830191533505902412, 6.24565089420628760623605522956, 6.55921701220043058458824166469, 6.87661178761912228051069824149, 7.987701642509487313713570245448
