| L(s) = 1 | − 1.41·2-s + (−2.13 + 2.10i)3-s + 2.00·4-s + (−2.73 − 4.18i)5-s + (3.02 − 2.97i)6-s − 4.11i·7-s − 2.82·8-s + (0.153 − 8.99i)9-s + (3.86 + 5.92i)10-s − 11.6i·11-s + (−4.27 + 4.20i)12-s − 7.53i·13-s + 5.82i·14-s + (14.6 + 3.21i)15-s + 4.00·16-s − 4.05·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.713 + 0.701i)3-s + 0.500·4-s + (−0.546 − 0.837i)5-s + (0.504 − 0.495i)6-s − 0.588i·7-s − 0.353·8-s + (0.0170 − 0.999i)9-s + (0.386 + 0.592i)10-s − 1.05i·11-s + (−0.356 + 0.350i)12-s − 0.579i·13-s + 0.416i·14-s + (0.976 + 0.214i)15-s + 0.250·16-s − 0.238·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1938714687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1938714687\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (2.13 - 2.10i)T \) |
| 5 | \( 1 + (2.73 + 4.18i)T \) |
| 23 | \( 1 + 4.79T \) |
| good | 7 | \( 1 + 4.11iT - 49T^{2} \) |
| 11 | \( 1 + 11.6iT - 121T^{2} \) |
| 13 | \( 1 + 7.53iT - 169T^{2} \) |
| 17 | \( 1 + 4.05T + 289T^{2} \) |
| 19 | \( 1 - 6.12T + 361T^{2} \) |
| 29 | \( 1 + 12.4iT - 841T^{2} \) |
| 31 | \( 1 + 7.25T + 961T^{2} \) |
| 37 | \( 1 - 16.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 40.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 14.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 5.66T + 2.80e3T^{2} \) |
| 59 | \( 1 + 8.63iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 13.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 24.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 139.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 68.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 35.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14.7iT - 9.40e3T^{2} \) |
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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
−9.860294222816501524017996186200, −8.963741959526739485074450404424, −8.253973738039251626306501516899, −7.31094680551259245137135195344, −6.14591551308497923593490599196, −5.29796750274607233267818611167, −4.21821942542877209228034507148, −3.25417752720982128830507636574, −1.07165174477305278363877866551, −0.11449018893528631729262665979,
1.70276604200974698865596910500, 2.71918216932079038792564825508, 4.33327974438459587214537626368, 5.61376085163919058556068488280, 6.60921593216485168308277587183, 7.20683102979328198786631419415, 7.907595357448232385324356624516, 8.983344083837185860519688362394, 9.985988238646011006996464474312, 10.79404969340115534831835799836
