L(s) = 1 | − 1.27·5-s + 2.45i·7-s − 4.60i·11-s − 5.38i·13-s − 2.30·17-s + (−0.0756 − 4.35i)19-s + 3.46i·23-s − 3.38·25-s + 4.46i·29-s − 0.666·31-s − 3.11i·35-s − 4.23i·37-s + 7.19i·41-s − 1.91i·43-s + 1.97i·47-s + ⋯ |
L(s) = 1 | − 0.568·5-s + 0.927i·7-s − 1.38i·11-s − 1.49i·13-s − 0.559·17-s + (−0.0173 − 0.999i)19-s + 0.723i·23-s − 0.677·25-s + 0.828i·29-s − 0.119·31-s − 0.526i·35-s − 0.696i·37-s + 1.12i·41-s − 0.291i·43-s + 0.288i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3020336411\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3020336411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.0756 + 4.35i)T \) |
good | 5 | \( 1 + 1.27T + 5T^{2} \) |
| 7 | \( 1 - 2.45iT - 7T^{2} \) |
| 11 | \( 1 + 4.60iT - 11T^{2} \) |
| 13 | \( 1 + 5.38iT - 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 4.46iT - 29T^{2} \) |
| 31 | \( 1 + 0.666T + 31T^{2} \) |
| 37 | \( 1 + 4.23iT - 37T^{2} \) |
| 41 | \( 1 - 7.19iT - 41T^{2} \) |
| 43 | \( 1 + 1.91iT - 43T^{2} \) |
| 47 | \( 1 - 1.97iT - 47T^{2} \) |
| 53 | \( 1 - 7.96iT - 53T^{2} \) |
| 59 | \( 1 + 0.672T + 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 + 3.25T + 67T^{2} \) |
| 71 | \( 1 + 5.48T + 71T^{2} \) |
| 73 | \( 1 + 5.78T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 7.51iT - 83T^{2} \) |
| 89 | \( 1 + 1.80iT - 89T^{2} \) |
| 97 | \( 1 - 6.69iT - 97T^{2} \) |
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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
−8.413823361685473654660539628398, −7.84895009636296279538517632777, −7.09783121928049411252442229862, −6.08534455149897765948165312590, −5.62421411611083969500637043890, −4.94172238456831761216640485178, −3.84660976336809384110467373282, −3.10980472308011682205500825050, −2.49004124971858625537777336262, −1.03210311103869255632598153879,
0.087524666159327572559204560735, 1.59108217441327106939271235852, 2.28494705770432372439551034081, 3.67434915165403740232594501462, 4.26987746392077235037882407766, 4.59990374869963485107384971033, 5.79629438468001237609979255993, 6.80972282409171293336546653466, 7.02973653591969208001970081942, 7.83696689173193273017515951667