| L(s) = 1 | − 5-s − 0.562·7-s + 6.36·11-s + 3.75·13-s − 6.65·17-s + 3.33·19-s + 4.16·23-s + 25-s + 9.82·29-s − 5.16·31-s + 0.562·35-s + 3.14·37-s + 11.5·41-s − 8.60·43-s − 12.2·47-s − 6.68·49-s + 0.434·53-s − 6.36·55-s − 1.73·59-s + 4.92·61-s − 3.75·65-s + 11.6·67-s + 11.7·71-s + 4.31·73-s − 3.58·77-s − 7.46·79-s − 8.99·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.212·7-s + 1.91·11-s + 1.04·13-s − 1.61·17-s + 0.765·19-s + 0.869·23-s + 0.200·25-s + 1.82·29-s − 0.928·31-s + 0.0951·35-s + 0.516·37-s + 1.80·41-s − 1.31·43-s − 1.79·47-s − 0.954·49-s + 0.0597·53-s − 0.858·55-s − 0.225·59-s + 0.630·61-s − 0.465·65-s + 1.42·67-s + 1.39·71-s + 0.505·73-s − 0.408·77-s − 0.839·79-s − 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171104082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171104082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 0.562T + 7T^{2} \) |
| 11 | \( 1 - 6.36T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 - 9.82T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 8.60T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 0.434T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + 8.99T + 83T^{2} \) |
| 89 | \( 1 + 6.36T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.236033002539456826822393049507, −7.05777347454397505681763027466, −6.63472435797889129936720316422, −6.17590989042800634201773962165, −5.02676542878284103842518043036, −4.28929082997473489712410051355, −3.68652490452323096534963332410, −2.91640515334168227146497924438, −1.65504576928254545675711145381, −0.815774673367089226528461573546,
0.815774673367089226528461573546, 1.65504576928254545675711145381, 2.91640515334168227146497924438, 3.68652490452323096534963332410, 4.28929082997473489712410051355, 5.02676542878284103842518043036, 6.17590989042800634201773962165, 6.63472435797889129936720316422, 7.05777347454397505681763027466, 8.236033002539456826822393049507
