| L(s) = 1 | − 0.684·2-s − 1.53·4-s + 1.50·5-s + 2.85·7-s + 2.41·8-s − 1.03·10-s + 11-s + 0.261·13-s − 1.95·14-s + 1.40·16-s + 3.17·17-s + 1.08·19-s − 2.30·20-s − 0.684·22-s − 7.64·23-s − 2.72·25-s − 0.179·26-s − 4.37·28-s − 9.16·29-s + 0.955·31-s − 5.79·32-s − 2.17·34-s + 4.30·35-s + 0.179·37-s − 0.745·38-s + 3.64·40-s − 1.82·41-s + ⋯ |
| L(s) = 1 | − 0.484·2-s − 0.765·4-s + 0.674·5-s + 1.08·7-s + 0.854·8-s − 0.326·10-s + 0.301·11-s + 0.0725·13-s − 0.522·14-s + 0.351·16-s + 0.768·17-s + 0.249·19-s − 0.516·20-s − 0.145·22-s − 1.59·23-s − 0.545·25-s − 0.0351·26-s − 0.826·28-s − 1.70·29-s + 0.171·31-s − 1.02·32-s − 0.372·34-s + 0.728·35-s + 0.0294·37-s − 0.120·38-s + 0.576·40-s − 0.284·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.684T + 2T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 - 2.85T + 7T^{2} \) |
| 13 | \( 1 - 0.261T + 13T^{2} \) |
| 17 | \( 1 - 3.17T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 + 9.16T + 29T^{2} \) |
| 31 | \( 1 - 0.955T + 31T^{2} \) |
| 37 | \( 1 - 0.179T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 3.10T + 59T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.988850407581720136557203739171, −7.28637318676257946798268893967, −6.13248271154491313259183884023, −5.57506384679295186368438048015, −4.85963081761752186130379616763, −4.13977303784724144093420562116, −3.31353504854134507219580308020, −1.82230932933189428325565419532, −1.51019013050397404491615700598, 0,
1.51019013050397404491615700598, 1.82230932933189428325565419532, 3.31353504854134507219580308020, 4.13977303784724144093420562116, 4.85963081761752186130379616763, 5.57506384679295186368438048015, 6.13248271154491313259183884023, 7.28637318676257946798268893967, 7.988850407581720136557203739171
