L(s) = 1 | + 2-s + 4-s + 5-s − 0.0653i·7-s + 8-s + 10-s − 1.77·11-s − 2.43i·13-s − 0.0653i·14-s + 16-s + 0.153i·17-s − 7.53·19-s + 20-s − 1.77·22-s − 2.47i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.0246i·7-s + 0.353·8-s + 0.316·10-s − 0.534·11-s − 0.674i·13-s − 0.0174i·14-s + 0.250·16-s + 0.0373i·17-s − 1.72·19-s + 0.223·20-s − 0.378·22-s − 0.515i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.560008516\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.560008516\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (1.89 + 7.96i)T \) |
good | 7 | \( 1 + 0.0653iT - 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 + 2.43iT - 13T^{2} \) |
| 17 | \( 1 - 0.153iT - 17T^{2} \) |
| 19 | \( 1 + 7.53T + 19T^{2} \) |
| 23 | \( 1 + 2.47iT - 23T^{2} \) |
| 29 | \( 1 + 4.70iT - 29T^{2} \) |
| 31 | \( 1 + 2.93iT - 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 + 6.63T + 41T^{2} \) |
| 43 | \( 1 - 4.71iT - 43T^{2} \) |
| 47 | \( 1 + 6.51iT - 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 0.249iT - 59T^{2} \) |
| 61 | \( 1 + 4.77iT - 61T^{2} \) |
| 71 | \( 1 - 0.869iT - 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 6.75iT - 79T^{2} \) |
| 83 | \( 1 - 0.191iT - 83T^{2} \) |
| 89 | \( 1 - 9.07iT - 89T^{2} \) |
| 97 | \( 1 - 5.89iT - 97T^{2} \) |
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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
−7.915474104225900242784218377135, −6.84070978935915461002623260339, −6.36666600770879225629627697390, −5.62696013045473736860589200485, −4.95460469283346343620680502223, −4.23090750953112644720630844784, −3.35952009977025895814825664842, −2.47818624666629770733707416820, −1.80077338856131601596808823197, −0.26347444865671712526870349967,
1.52315504509131316662443380892, 2.23736734164903071543412130172, 3.15320255194069379753266006840, 4.00029737204237209634684493320, 4.78313327042302380737565804169, 5.39721799190229922091321853436, 6.18289666653137210664846857356, 6.80744741461655020663191318599, 7.39077717992711589613183674975, 8.441312798722624079594199984381