| L(s) = 1 | + (0.990 + 0.139i)2-s + (0.530 − 2.12i)3-s + (0.961 + 0.275i)4-s + (0.0547 − 1.56i)5-s + (0.821 − 2.03i)6-s + (2.87 − 1.27i)7-s + (0.913 + 0.406i)8-s + (−1.59 − 0.847i)9-s + (0.272 − 1.54i)10-s + (−1.49 + 2.95i)11-s + (1.09 − 1.89i)12-s + (−2.34 + 1.46i)13-s + (3.02 − 0.866i)14-s + (−3.30 − 0.947i)15-s + (0.848 + 0.529i)16-s + (−6.27 + 3.33i)17-s + ⋯ |
| L(s) = 1 | + (0.700 + 0.0984i)2-s + (0.306 − 1.22i)3-s + (0.480 + 0.137i)4-s + (0.0244 − 0.700i)5-s + (0.335 − 0.829i)6-s + (1.08 − 0.483i)7-s + (0.322 + 0.143i)8-s + (−0.531 − 0.282i)9-s + (0.0860 − 0.488i)10-s + (−0.451 + 0.892i)11-s + (0.316 − 0.548i)12-s + (−0.649 + 0.405i)13-s + (0.807 − 0.231i)14-s + (−0.852 − 0.244i)15-s + (0.212 + 0.132i)16-s + (−1.52 + 0.809i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94830 - 1.39446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94830 - 1.39446i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.990 - 0.139i)T \) |
| 11 | \( 1 + (1.49 - 2.95i)T \) |
| 19 | \( 1 + (-2.16 + 3.78i)T \) |
| good | 3 | \( 1 + (-0.530 + 2.12i)T + (-2.64 - 1.40i)T^{2} \) |
| 5 | \( 1 + (-0.0547 + 1.56i)T + (-4.98 - 0.348i)T^{2} \) |
| 7 | \( 1 + (-2.87 + 1.27i)T + (4.68 - 5.20i)T^{2} \) |
| 13 | \( 1 + (2.34 - 1.46i)T + (5.69 - 11.6i)T^{2} \) |
| 17 | \( 1 + (6.27 - 3.33i)T + (9.50 - 14.0i)T^{2} \) |
| 23 | \( 1 + (1.39 - 0.506i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.22 + 1.18i)T + (1.01 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-3.78 - 4.20i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-2.62 - 1.90i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.410 + 1.64i)T + (-36.2 - 19.2i)T^{2} \) |
| 43 | \( 1 + (11.8 + 4.32i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-3.83 - 7.85i)T + (-28.9 + 37.0i)T^{2} \) |
| 53 | \( 1 + (-0.0190 - 0.544i)T + (-52.8 + 3.69i)T^{2} \) |
| 59 | \( 1 + (2.51 - 5.16i)T + (-36.3 - 46.4i)T^{2} \) |
| 61 | \( 1 + (4.73 + 6.05i)T + (-14.7 + 59.1i)T^{2} \) |
| 67 | \( 1 + (0.957 - 5.42i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.319 - 9.14i)T + (-70.8 - 4.95i)T^{2} \) |
| 73 | \( 1 + (-6.81 + 0.476i)T + (72.2 - 10.1i)T^{2} \) |
| 79 | \( 1 + (2.27 + 5.63i)T + (-56.8 + 54.8i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 2.24i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (6.95 + 5.83i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-16.6 - 2.33i)T + (93.2 + 26.7i)T^{2} \) |
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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
−11.35972242502466821918925405847, −10.30620894601467477057742125343, −8.873687348842620200303612832844, −8.018470839769655041901543843717, −7.24652964106173141331279969458, −6.52877500940092537176835232634, −4.91375978826288785812493035497, −4.48326757900889289687099990834, −2.44455400785055821521072380638, −1.48467935381645942622578882971,
2.42072486681924776525417625634, 3.37434828812036282859515179608, 4.62721728187277075072644116581, 5.23264313870099891503204802113, 6.47430520237879818240967849622, 7.77480390482131378903734532441, 8.701630523162824643376733261779, 9.798313101509532633808685096719, 10.63690725833082278186942984010, 11.24534838653494110731016670462
