| L(s) = 1 | + 5·2-s + 6·3-s − 7·4-s − 25·5-s + 30·6-s − 244·7-s − 195·8-s − 207·9-s − 125·10-s + 794·11-s − 42·12-s − 169·13-s − 1.22e3·14-s − 150·15-s − 751·16-s − 1.53e3·17-s − 1.03e3·18-s + 2.70e3·19-s + 175·20-s − 1.46e3·21-s + 3.97e3·22-s − 702·23-s − 1.17e3·24-s + 625·25-s − 845·26-s − 2.70e3·27-s + 1.70e3·28-s + ⋯ |
| L(s) = 1 | + 0.883·2-s + 0.384·3-s − 0.218·4-s − 0.447·5-s + 0.340·6-s − 1.88·7-s − 1.07·8-s − 0.851·9-s − 0.395·10-s + 1.97·11-s − 0.0841·12-s − 0.277·13-s − 1.66·14-s − 0.172·15-s − 0.733·16-s − 1.28·17-s − 0.752·18-s + 1.71·19-s + 0.0978·20-s − 0.724·21-s + 1.74·22-s − 0.276·23-s − 0.414·24-s + 1/5·25-s − 0.245·26-s − 0.712·27-s + 0.411·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - 5 T + p^{5} T^{2} \) |
| 3 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 244 T + p^{5} T^{2} \) |
| 11 | \( 1 - 794 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1534 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2706 T + p^{5} T^{2} \) |
| 23 | \( 1 + 702 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5038 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3634 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7058 T + p^{5} T^{2} \) |
| 41 | \( 1 + 294 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7618 T + p^{5} T^{2} \) |
| 47 | \( 1 + 3020 T + p^{5} T^{2} \) |
| 53 | \( 1 - 626 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30066 T + p^{5} T^{2} \) |
| 61 | \( 1 + 5806 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12436 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4734 T + p^{5} T^{2} \) |
| 73 | \( 1 + 14694 T + p^{5} T^{2} \) |
| 79 | \( 1 + 39804 T + p^{5} T^{2} \) |
| 83 | \( 1 + 41776 T + p^{5} T^{2} \) |
| 89 | \( 1 - 7970 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78050 T + p^{5} T^{2} \) |
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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
−13.48930391804837347601379937064, −12.38030236217803321319551685259, −11.54688191608099333040674511666, −9.401256699229021353461743707036, −9.059771108121277185617321513720, −6.92110022865449866282603655496, −5.84462159905689840514688138944, −3.93818613014492224450947705413, −3.15080588148727670546776940961, 0,
3.15080588148727670546776940961, 3.93818613014492224450947705413, 5.84462159905689840514688138944, 6.92110022865449866282603655496, 9.059771108121277185617321513720, 9.401256699229021353461743707036, 11.54688191608099333040674511666, 12.38030236217803321319551685259, 13.48930391804837347601379937064
