| L(s) = 1 | − 2.68·3-s − 1.86·5-s − 7-s + 4.22·9-s + 0.846·11-s + 2.55·13-s + 5.00·15-s − 7.07·17-s + 0.476·19-s + 2.68·21-s + 23-s − 1.53·25-s − 3.30·27-s + 8.63·29-s − 3.31·31-s − 2.27·33-s + 1.86·35-s + 7.85·37-s − 6.85·39-s + 2.82·41-s + 0.274·43-s − 7.87·45-s + 13.4·47-s + 49-s + 19.0·51-s + 8.93·53-s − 1.57·55-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 0.832·5-s − 0.377·7-s + 1.40·9-s + 0.255·11-s + 0.707·13-s + 1.29·15-s − 1.71·17-s + 0.109·19-s + 0.586·21-s + 0.208·23-s − 0.306·25-s − 0.635·27-s + 1.60·29-s − 0.594·31-s − 0.396·33-s + 0.314·35-s + 1.29·37-s − 1.09·39-s + 0.441·41-s + 0.0419·43-s − 1.17·45-s + 1.96·47-s + 0.142·49-s + 2.66·51-s + 1.22·53-s − 0.212·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 + 1.86T + 5T^{2} \) |
| 11 | \( 1 - 0.846T + 11T^{2} \) |
| 13 | \( 1 - 2.55T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 0.476T + 19T^{2} \) |
| 29 | \( 1 - 8.63T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 0.274T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 - 8.93T + 53T^{2} \) |
| 59 | \( 1 + 1.66T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 7.31T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.72T + 83T^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−8.559489845634659233388716363817, −7.53173043701205727533076696006, −6.77800482136463711638528280653, −6.22550627246278159256783099908, −5.50906612457544477269276395043, −4.39089215280688668434137673731, −4.09350911448998470927197000474, −2.67568458025126800538962654152, −1.08319853328756332988166989997, 0,
1.08319853328756332988166989997, 2.67568458025126800538962654152, 4.09350911448998470927197000474, 4.39089215280688668434137673731, 5.50906612457544477269276395043, 6.22550627246278159256783099908, 6.77800482136463711638528280653, 7.53173043701205727533076696006, 8.559489845634659233388716363817
