| L(s) = 1 | − 2.73·3-s + 5-s − 4.73·7-s + 4.46·9-s + 3.46·11-s + 3.46·13-s − 2.73·15-s − 3.46·17-s + 2·19-s + 12.9·21-s + 2.19·23-s + 25-s − 3.99·27-s − 2.53·31-s − 9.46·33-s − 4.73·35-s − 6·37-s − 9.46·39-s − 9.46·41-s − 0.196·43-s + 4.46·45-s + 2.19·47-s + 15.3·49-s + 9.46·51-s − 10.3·53-s + 3.46·55-s − 5.46·57-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.447·5-s − 1.78·7-s + 1.48·9-s + 1.04·11-s + 0.960·13-s − 0.705·15-s − 0.840·17-s + 0.458·19-s + 2.82·21-s + 0.457·23-s + 0.200·25-s − 0.769·27-s − 0.455·31-s − 1.64·33-s − 0.799·35-s − 0.986·37-s − 1.51·39-s − 1.47·41-s − 0.0299·43-s + 0.665·45-s + 0.320·47-s + 2.19·49-s + 1.32·51-s − 1.42·53-s + 0.467·55-s − 0.723·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 0.928T + 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.369617944511913742317556790155, −8.788555865680087591935935879703, −7.05888379108100852942441508436, −6.57444068311977131761921957332, −6.08829428817590354920690143168, −5.28995590108713009777018455637, −4.10912843233860059316949401395, −3.17331369713556042964900992061, −1.38644316172211549871292105859, 0,
1.38644316172211549871292105859, 3.17331369713556042964900992061, 4.10912843233860059316949401395, 5.28995590108713009777018455637, 6.08829428817590354920690143168, 6.57444068311977131761921957332, 7.05888379108100852942441508436, 8.788555865680087591935935879703, 9.369617944511913742317556790155
