L(s) = 1 | + 2-s + (1.29 + 1.15i)3-s + 4-s + (2.22 − 0.223i)5-s + (1.29 + 1.15i)6-s + 0.666·7-s + 8-s + (0.348 + 2.97i)9-s + (2.22 − 0.223i)10-s − 4.92·11-s + (1.29 + 1.15i)12-s + 3.71i·13-s + 0.666·14-s + (3.13 + 2.27i)15-s + 16-s + 0.810i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.747 + 0.664i)3-s + 0.5·4-s + (0.994 − 0.100i)5-s + (0.528 + 0.470i)6-s + 0.252·7-s + 0.353·8-s + (0.116 + 0.993i)9-s + (0.703 − 0.0707i)10-s − 1.48·11-s + (0.373 + 0.332i)12-s + 1.02i·13-s + 0.178·14-s + (0.809 + 0.586i)15-s + 0.250·16-s + 0.196i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.06161 + 1.12195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.06161 + 1.12195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.29 - 1.15i)T \) |
| 5 | \( 1 + (-2.22 + 0.223i)T \) |
| 23 | \( 1 + (-1.14 + 4.65i)T \) |
good | 7 | \( 1 - 0.666T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 3.71iT - 13T^{2} \) |
| 17 | \( 1 - 0.810iT - 17T^{2} \) |
| 19 | \( 1 + 4.16iT - 19T^{2} \) |
| 29 | \( 1 + 9.02iT - 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 - 2.21T + 37T^{2} \) |
| 41 | \( 1 + 2.54iT - 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 6.81iT - 59T^{2} \) |
| 61 | \( 1 - 9.21iT - 61T^{2} \) |
| 67 | \( 1 - 7.95T + 67T^{2} \) |
| 71 | \( 1 + 14.5iT - 71T^{2} \) |
| 73 | \( 1 - 4.84iT - 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + 13.5iT - 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
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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
−10.60130690751548360426634693156, −9.731785919657229767070251495143, −8.954861449840770056520474236231, −8.013221635312278810627415909963, −7.00337157412684856907432583490, −5.82472599447944056299184609548, −4.96295905053613237556771457926, −4.23089145957614615228953282004, −2.74737580494079417874047899477, −2.12271639154156774474151887198,
1.57169655925906509253625244129, 2.66468652415207796521437715290, 3.47354253055052401470554952475, 5.21774422813498749398078935048, 5.66326796715567028950546241867, 6.88088917029772463279090417750, 7.68228030780507333805981418351, 8.442760909108210435325089329723, 9.601680191474309572943790387660, 10.37516403358174658255797037606