| L(s) = 1 | − i·2-s − i·3-s + 4-s − 6-s − 2i·7-s − 3i·8-s − 9-s − 6·11-s − i·12-s − 2·14-s − 16-s − 6i·17-s + i·18-s − 19-s − 2·21-s + 6i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s + 0.5·4-s − 0.408·6-s − 0.755i·7-s − 1.06i·8-s − 0.333·9-s − 1.80·11-s − 0.288i·12-s − 0.534·14-s − 0.250·16-s − 1.45i·17-s + 0.235i·18-s − 0.229·19-s − 0.436·21-s + 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130785517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130785517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.351471951766441361473361864459, −7.982338082391660254885536409864, −7.46515482458678458890735538015, −6.89114476296741186754401743500, −5.73321900409454189265297261868, −4.88347127068724331216009089231, −3.51153831935527777392753469217, −2.75537441579377559796921450211, −1.75986914925693654187063371334, −0.40487527694469688881034360798,
2.17363756794423100136012347804, 2.84084566744910887499582391347, 4.25081530838063350370727613507, 5.32341880375655928565008582686, 5.79072266519328964800501616243, 6.63237319163602887396808023249, 7.75331431266845502753519158350, 8.267226605916544117749299181595, 8.947217501679964739528366579863, 10.15479965577495699832978911352
