L(s) = 1 | − 2.90i·3-s + (0.311 + 2.21i)5-s + 3.52i·7-s − 5.42·9-s + 3.80·11-s + 2.62i·13-s + (6.42 − 0.903i)15-s + 5.80i·17-s + 5.05·19-s + 10.2·21-s − 0.474i·23-s + (−4.80 + 1.37i)25-s + 7.05i·27-s − 2·29-s − 2.75·31-s + ⋯ |
L(s) = 1 | − 1.67i·3-s + (0.139 + 0.990i)5-s + 1.33i·7-s − 1.80·9-s + 1.14·11-s + 0.727i·13-s + (1.65 − 0.233i)15-s + 1.40i·17-s + 1.15·19-s + 2.23·21-s − 0.0989i·23-s + (−0.961 + 0.275i)25-s + 1.35i·27-s − 0.371·29-s − 0.494·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50215 + 0.105009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50215 + 0.105009i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.311 - 2.21i)T \) |
good | 3 | \( 1 + 2.90iT - 3T^{2} \) |
| 7 | \( 1 - 3.52iT - 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 - 5.80iT - 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 + 0.474iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 7.18iT - 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 1.95iT - 43T^{2} \) |
| 47 | \( 1 + 5.33iT - 47T^{2} \) |
| 53 | \( 1 - 5.37iT - 53T^{2} \) |
| 59 | \( 1 + 5.05T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.76iT - 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 - 6.66iT - 73T^{2} \) |
| 79 | \( 1 - 5.24T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−10.91100721911104003899166123092, −9.478426027457014519264847065592, −8.783798161513166031194311813743, −7.76842796206476162647302307070, −6.96881599148936617551710609413, −6.21817837838549063003196842060, −5.68284981720284476121226986336, −3.67944701913127203620477790325, −2.41088061620223998540545979992, −1.62078477626854186280736772918,
0.901204714652685715508294568582, 3.26488095340401798146517027395, 4.09564573872738959995880026514, 4.84145966415463868409578408818, 5.62110888284490855133152463288, 7.07614367087467986560794737074, 8.122497301501324793317918745053, 9.295154003574856759065873855141, 9.549598531832885635262398119834, 10.36105697935487621119699950495