| L(s) = 1 | + (0.156 − 0.987i)2-s + (−0.0340 + 0.00539i)3-s + (−0.951 − 0.309i)4-s + (0.0723 + 2.23i)5-s + 0.0345i·6-s + (0.886 + 1.74i)7-s + (−0.453 + 0.891i)8-s + (−2.85 + 0.926i)9-s + (2.21 + 0.278i)10-s + (5.39 + 1.75i)11-s + (0.0340 + 0.00539i)12-s + (2.98 − 0.472i)13-s + (1.85 − 0.603i)14-s + (−0.0145 − 0.0757i)15-s + (0.809 + 0.587i)16-s + (−1.26 + 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.110 − 0.698i)2-s + (−0.0196 + 0.00311i)3-s + (−0.475 − 0.154i)4-s + (0.0323 + 0.999i)5-s + 0.0140i·6-s + (0.335 + 0.657i)7-s + (−0.160 + 0.315i)8-s + (−0.950 + 0.308i)9-s + (0.701 + 0.0879i)10-s + (1.62 + 0.528i)11-s + (0.00984 + 0.00155i)12-s + (0.827 − 0.131i)13-s + (0.496 − 0.161i)14-s + (−0.00375 − 0.0195i)15-s + (0.202 + 0.146i)16-s + (−0.306 + 0.602i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32030 + 0.181482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32030 + 0.181482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 + 0.987i)T \) |
| 5 | \( 1 + (-0.0723 - 2.23i)T \) |
| 31 | \( 1 + (5.31 - 1.64i)T \) |
| good | 3 | \( 1 + (0.0340 - 0.00539i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.886 - 1.74i)T + (-4.11 + 5.66i)T^{2} \) |
| 11 | \( 1 + (-5.39 - 1.75i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.98 + 0.472i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.26 - 2.48i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.11 + 1.53i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0394 + 0.0200i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-7.51 + 5.46i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-2.60 - 2.60i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.10 + 2.98i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (0.816 - 5.15i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (2.48 - 0.394i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (10.9 + 5.59i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.64 + 2.26i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + 6.54iT - 61T^{2} \) |
| 67 | \( 1 + (7.51 - 7.51i)T - 67iT^{2} \) |
| 71 | \( 1 + (-0.478 - 1.47i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.59 - 7.04i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (0.636 + 1.95i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.509 + 3.21i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.49 + 13.8i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (5.47 + 10.7i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.43645548095461668976912926758, −11.15983297311029360649111923758, −9.993594471582952307060071619657, −8.980119924460921514326044935508, −8.180744335386029030550034613511, −6.62466002462660134490790253629, −5.87537378541828591778097389539, −4.35719941458264066581693005101, −3.16646108851282724772472058131, −1.92747096272342269221457362993,
1.07555717126524769855056107361, 3.61669220423994697555931288301, 4.57298871419264402389752555023, 5.82076737170474267258832635846, 6.60952676728809150673306194366, 7.928610548161511376972425083550, 8.878949072781139297478519387155, 9.238020927449362457491458972385, 10.86456405215620252731391393828, 11.73295859233970207035267115401
