| L(s) = 1 | + (0.212 − 0.977i)2-s + (0.800 − 0.599i)3-s + (−0.909 − 0.415i)4-s + (−1.43 + 1.71i)5-s + (−0.415 − 0.909i)6-s + (0.00180 − 0.0252i)7-s + (−0.599 + 0.800i)8-s + (0.281 − 0.959i)9-s + (1.37 + 1.76i)10-s + (1.53 − 2.38i)11-s + (−0.977 + 0.212i)12-s + (4.52 − 0.323i)13-s + (−0.0242 − 0.00712i)14-s + (−0.120 + 2.23i)15-s + (0.654 + 0.755i)16-s + (1.95 − 5.23i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.690i)2-s + (0.462 − 0.345i)3-s + (−0.454 − 0.207i)4-s + (−0.641 + 0.767i)5-s + (−0.169 − 0.371i)6-s + (0.000682 − 0.00954i)7-s + (−0.211 + 0.283i)8-s + (0.0939 − 0.319i)9-s + (0.433 + 0.558i)10-s + (0.462 − 0.719i)11-s + (−0.282 + 0.0613i)12-s + (1.25 − 0.0896i)13-s + (−0.00648 − 0.00190i)14-s + (−0.0309 + 0.576i)15-s + (0.163 + 0.188i)16-s + (0.473 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0920 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0920 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11922 - 1.22740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11922 - 1.22740i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 3 | \( 1 + (-0.800 + 0.599i)T \) |
| 5 | \( 1 + (1.43 - 1.71i)T \) |
| 23 | \( 1 + (2.38 + 4.15i)T \) |
| good | 7 | \( 1 + (-0.00180 + 0.0252i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-1.53 + 2.38i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.52 + 0.323i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-1.95 + 5.23i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.38 + 3.03i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (1.38 - 0.632i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.359 - 2.50i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.84 + 3.37i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-3.75 + 1.10i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-4.72 - 6.30i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (7.10 - 7.10i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.70 - 0.550i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (3.34 + 2.89i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (4.38 - 0.630i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-14.4 - 3.13i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (13.6 - 8.80i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (3.27 - 1.22i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-2.17 + 2.50i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.90 + 1.58i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-1.29 + 8.99i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.05 - 3.30i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.45857642827085589880431132724, −9.336732101597991244261397951094, −8.610630428188957739537048428780, −7.70481130739853869791780671077, −6.75779647675671371667754587906, −5.80324428535394948652905630667, −4.32993489448956552891739378170, −3.39142683103920856384015421677, −2.64841918938814499796910980458, −0.911036890750437008549884785356,
1.54019113762491115785501831131, 3.72426287720341890051024462108, 4.00331418649326282702293753970, 5.29458898563246944754146499932, 6.16304229567048664541231828227, 7.41749510113214149036604579577, 8.125064211215245246313962545281, 8.782468884322277948447909603029, 9.594415225530905962441538430121, 10.53439594426627155235757578247
