| L(s) = 1 | − 10·2-s + 14·3-s + 68·4-s + 56·5-s − 140·6-s − 360·8-s − 47·9-s − 560·10-s + 232·11-s + 952·12-s + 140·13-s + 784·15-s + 1.42e3·16-s + 1.72e3·17-s + 470·18-s + 98·19-s + 3.80e3·20-s − 2.32e3·22-s + 1.82e3·23-s − 5.04e3·24-s + 11·25-s − 1.40e3·26-s − 4.06e3·27-s + 3.41e3·29-s − 7.84e3·30-s + 7.64e3·31-s − 2.72e3·32-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.898·3-s + 17/8·4-s + 1.00·5-s − 1.58·6-s − 1.98·8-s − 0.193·9-s − 1.77·10-s + 0.578·11-s + 1.90·12-s + 0.229·13-s + 0.899·15-s + 1.39·16-s + 1.44·17-s + 0.341·18-s + 0.0622·19-s + 2.12·20-s − 1.02·22-s + 0.718·23-s − 1.78·24-s + 0.00351·25-s − 0.406·26-s − 1.07·27-s + 0.754·29-s − 1.59·30-s + 1.42·31-s − 0.469·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.196461872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196461872\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 5 | \( 1 - 56 T + p^{5} T^{2} \) |
| 11 | \( 1 - 232 T + p^{5} T^{2} \) |
| 13 | \( 1 - 140 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1722 T + p^{5} T^{2} \) |
| 19 | \( 1 - 98 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1824 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3418 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7644 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10398 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17962 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10880 T + p^{5} T^{2} \) |
| 47 | \( 1 + 9324 T + p^{5} T^{2} \) |
| 53 | \( 1 - 2262 T + p^{5} T^{2} \) |
| 59 | \( 1 - 2730 T + p^{5} T^{2} \) |
| 61 | \( 1 + 25648 T + p^{5} T^{2} \) |
| 67 | \( 1 + 48404 T + p^{5} T^{2} \) |
| 71 | \( 1 + 58560 T + p^{5} T^{2} \) |
| 73 | \( 1 + 68082 T + p^{5} T^{2} \) |
| 79 | \( 1 - 31784 T + p^{5} T^{2} \) |
| 83 | \( 1 - 20538 T + p^{5} T^{2} \) |
| 89 | \( 1 - 50582 T + p^{5} T^{2} \) |
| 97 | \( 1 - 58506 T + p^{5} T^{2} \) |
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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
−14.69346113629062205792820335152, −13.71783649049906696937209348151, −11.91810441229952272733970212295, −10.45889534209058318498975779880, −9.520257264082181074499874446997, −8.739051791920234788535013696234, −7.61520329205596700571720150594, −6.09450485603994223222378877207, −2.80855483746860376476991668666, −1.30752165276660555124216235380,
1.30752165276660555124216235380, 2.80855483746860376476991668666, 6.09450485603994223222378877207, 7.61520329205596700571720150594, 8.739051791920234788535013696234, 9.520257264082181074499874446997, 10.45889534209058318498975779880, 11.91810441229952272733970212295, 13.71783649049906696937209348151, 14.69346113629062205792820335152
