| L(s) = 1 | + 3-s + 5-s − 1.14·7-s + 9-s + 5.14·11-s + 13-s + 15-s − 3.86·17-s + 0.726·19-s − 1.14·21-s + 0.414·23-s + 25-s + 27-s + 9.73·29-s − 3.00·31-s + 5.14·33-s − 1.14·35-s + 3.86·37-s + 39-s − 3.86·41-s − 6.28·43-s + 45-s + 5.55·47-s − 5.69·49-s − 3.86·51-s + 0.132·53-s + 5.14·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.431·7-s + 0.333·9-s + 1.55·11-s + 0.277·13-s + 0.258·15-s − 0.938·17-s + 0.166·19-s − 0.249·21-s + 0.0864·23-s + 0.200·25-s + 0.192·27-s + 1.80·29-s − 0.540·31-s + 0.894·33-s − 0.192·35-s + 0.635·37-s + 0.160·39-s − 0.604·41-s − 0.958·43-s + 0.149·45-s + 0.810·47-s − 0.813·49-s − 0.541·51-s + 0.0181·53-s + 0.693·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.026501386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.026501386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 - 0.726T + 19T^{2} \) |
| 23 | \( 1 - 0.414T + 23T^{2} \) |
| 29 | \( 1 - 9.73T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 3.86T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 5.55T + 47T^{2} \) |
| 53 | \( 1 - 0.132T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 7.00T + 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 + 8.46T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 8.69T + 97T^{2} \) |
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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
−8.364236018991018050419860523789, −7.11661089126653387406219146906, −6.68173123621137054472554063716, −6.15453481456826253262807355734, −5.12977231017379122042187909187, −4.26425080728962727789934203897, −3.63881912401998310164892564048, −2.76946111654194041444656308308, −1.87725245895059381738622996163, −0.922495400023116751151042188441,
0.922495400023116751151042188441, 1.87725245895059381738622996163, 2.76946111654194041444656308308, 3.63881912401998310164892564048, 4.26425080728962727789934203897, 5.12977231017379122042187909187, 6.15453481456826253262807355734, 6.68173123621137054472554063716, 7.11661089126653387406219146906, 8.364236018991018050419860523789
