| L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.868 − 1.00i)3-s + (0.841 + 0.540i)4-s + (−0.397 + 2.76i)5-s + (−1.11 + 0.716i)6-s + (−0.415 − 0.909i)7-s + (−0.654 − 0.755i)8-s + (0.176 + 1.22i)9-s + (1.15 − 2.53i)10-s + (5.62 − 1.65i)11-s + (1.27 − 0.373i)12-s + (−2.66 + 5.82i)13-s + (0.142 + 0.989i)14-s + (2.42 + 2.79i)15-s + (0.415 + 0.909i)16-s + (−0.167 + 0.107i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.501 − 0.578i)3-s + (0.420 + 0.270i)4-s + (−0.177 + 1.23i)5-s + (−0.455 + 0.292i)6-s + (−0.157 − 0.343i)7-s + (−0.231 − 0.267i)8-s + (0.0589 + 0.409i)9-s + (0.366 − 0.803i)10-s + (1.69 − 0.498i)11-s + (0.367 − 0.107i)12-s + (−0.738 + 1.61i)13-s + (0.0380 + 0.264i)14-s + (0.625 + 0.722i)15-s + (0.103 + 0.227i)16-s + (−0.0405 + 0.0260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12263 + 0.194997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12263 + 0.194997i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (1.51 + 4.55i)T \) |
| good | 3 | \( 1 + (-0.868 + 1.00i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.397 - 2.76i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-5.62 + 1.65i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.66 - 5.82i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.167 - 0.107i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.30 - 2.12i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.01 + 2.57i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.141 - 0.163i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.164 - 1.14i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.668 + 4.65i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (1.00 - 1.16i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 6.79T + 47T^{2} \) |
| 53 | \( 1 + (-2.78 - 6.10i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-4.10 + 8.97i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (3.44 + 3.98i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (10.4 + 3.05i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (14.6 + 4.31i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.06 - 3.89i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-2.79 + 6.11i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.912 - 6.34i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (0.409 - 0.472i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.02 - 7.13i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.62125146274471315958836791730, −10.74086609956371596353259080233, −9.770095325384004999088005546643, −8.889572746175678239638777181319, −7.80439798073941476747291551305, −6.87175428761885265210788030909, −6.49598416071666921201002530422, −4.21252349779705502440258850787, −2.97707242653988523765690401453, −1.71829608234633112308209150405,
1.11112100273304965018590145252, 3.15723738766216343999337603479, 4.48016530996307526425717222286, 5.58453515598607831720111092006, 6.91955418546317881342821134926, 8.080493243990641445967094184606, 8.960051516921105480102607756772, 9.465724492068913111926636320872, 10.18521734600598365293366418831, 11.78756906153897692485197487565
