| L(s) = 1 | + i·3-s − i·7-s − 9-s + i·13-s − 3·19-s + 21-s + 4i·23-s − i·27-s − 4·29-s − 7·31-s + 6i·37-s − 39-s + 6·41-s + 9i·43-s − 6i·47-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.277i·13-s − 0.688·19-s + 0.218·21-s + 0.834i·23-s − 0.192i·27-s − 0.742·29-s − 1.25·31-s + 0.986i·37-s − 0.160·39-s + 0.937·41-s + 1.37i·43-s − 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7559420591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7559420591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−9.340910973639727101030373707159, −8.639769256121958995173049553935, −7.74570039723333436377873277899, −7.05054065531410228618352597696, −6.08917022798132104600834804322, −5.35210039435203600765445605149, −4.38214605879458415108493691353, −3.76211544891827657081617539049, −2.72292101756700228693606410788, −1.47272770460912329782683505230,
0.24592817794643946973030405775, 1.75387266958191001665915908677, 2.60224697650086629887193494503, 3.68823231709150720911723401967, 4.67419065467113868430514364543, 5.72223677903823098917009684268, 6.18381423301317560512408295484, 7.27930259799018092915697048342, 7.68276768189239902281564746574, 8.869210241208034633707728558690
