L(s) = 1 | + 0.0838i·2-s + i·3-s + 1.99·4-s + (−2.20 − 0.366i)5-s − 0.0838·6-s − 2.34i·7-s + 0.334i·8-s − 9-s + (0.0307 − 0.184i)10-s + 1.99i·12-s + 5.61i·13-s + 0.196·14-s + (0.366 − 2.20i)15-s + 3.95·16-s − 2.13i·17-s − 0.0838i·18-s + ⋯ |
L(s) = 1 | + 0.0592i·2-s + 0.577i·3-s + 0.996·4-s + (−0.986 − 0.164i)5-s − 0.0342·6-s − 0.885i·7-s + 0.118i·8-s − 0.333·9-s + (0.00972 − 0.0584i)10-s + 0.575i·12-s + 1.55i·13-s + 0.0524·14-s + (0.0947 − 0.569i)15-s + 0.989·16-s − 0.518i·17-s − 0.0197i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.615955733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.615955733\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.20 + 0.366i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.0838iT - 2T^{2} \) |
| 7 | \( 1 + 2.34iT - 7T^{2} \) |
| 13 | \( 1 - 5.61iT - 13T^{2} \) |
| 17 | \( 1 + 2.13iT - 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 8.11iT - 23T^{2} \) |
| 29 | \( 1 + 5.11T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 5.25iT - 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 6.99iT - 43T^{2} \) |
| 47 | \( 1 - 5.75iT - 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 9.16iT - 67T^{2} \) |
| 71 | \( 1 + 6.51T + 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 + 11.7iT - 83T^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 + 2.10iT - 97T^{2} \) |
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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
−9.464886263105733680037286516498, −8.663150538036415025411354300741, −7.59438280391082942992705461775, −7.24940039779601522930962745290, −6.41910699304454843927327667702, −5.33689748491900074601655428611, −4.22122789680381463360240609858, −3.80605886070754005653379876843, −2.63830251457270418657187419128, −1.26020688724189536521142677271,
0.62928612602977422869055606578, 2.22914459846481963663492636260, 2.86731775884667050250234375920, 3.86562159142047537064247161768, 5.20289307807273443557069956263, 6.07636702662339320711661648507, 6.67497877873341867600660073151, 7.64703144706164113382564966888, 8.122913650296352549199452728184, 8.757806760384298615980468914377