L(s) = 1 | − i·2-s − 4-s − 4i·7-s + i·8-s − i·13-s − 4·14-s + 16-s − 2i·17-s − 4·19-s − 8i·23-s − 26-s + 4i·28-s + 2·29-s − 8·31-s − i·32-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.51i·7-s + 0.353i·8-s − 0.277i·13-s − 1.06·14-s + 0.250·16-s − 0.485i·17-s − 0.917·19-s − 1.66i·23-s − 0.196·26-s + 0.755i·28-s + 0.371·29-s − 1.43·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6768692424\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6768692424\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−7.76196405449442076734416771721, −6.89084006437146102587944905895, −6.34512414502128196199245344892, −5.23686293210021983107872850964, −4.43793212707552996690108082970, −3.99349061079054980620793294183, −3.08044305662961324279902293412, −2.18883706441261015244961545066, −1.06092313251401736115090796316, −0.18852261676169983550712432673,
1.60092374720150283550564529370, 2.44541812049859460712219601008, 3.50380977256844395006186230173, 4.28631455254550334423450347841, 5.34517030754518687892604002487, 5.65057238785939541610352624124, 6.37891583083903152321327954100, 7.13918252407822289133307190687, 7.919988073666175394694549378834, 8.534152820672841764531475712962