| L(s) = 1 | + 0.608·2-s − 1.62·4-s + 1.59·5-s − 1.08·7-s − 2.20·8-s + 0.972·10-s + 11-s + 1.21·13-s − 0.661·14-s + 1.91·16-s − 1.83·17-s + 6.08·19-s − 2.60·20-s + 0.608·22-s − 8.06·23-s − 2.44·25-s + 0.737·26-s + 1.77·28-s + 4.42·29-s − 8.64·31-s + 5.58·32-s − 1.11·34-s − 1.73·35-s + 5.08·37-s + 3.70·38-s − 3.52·40-s − 9.86·41-s + ⋯ |
| L(s) = 1 | + 0.430·2-s − 0.814·4-s + 0.714·5-s − 0.411·7-s − 0.780·8-s + 0.307·10-s + 0.301·11-s + 0.336·13-s − 0.176·14-s + 0.479·16-s − 0.444·17-s + 1.39·19-s − 0.582·20-s + 0.129·22-s − 1.68·23-s − 0.488·25-s + 0.144·26-s + 0.335·28-s + 0.822·29-s − 1.55·31-s + 0.986·32-s − 0.191·34-s − 0.293·35-s + 0.835·37-s + 0.600·38-s − 0.558·40-s − 1.54·41-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 | \( 1 - 0.608T + 2T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 + 8.06T + 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 - 5.08T + 37T^{2} \) |
| 41 | \( 1 + 9.86T + 41T^{2} \) |
| 43 | \( 1 + 2.74T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6.40T + 73T^{2} \) |
| 79 | \( 1 - 4.44T + 79T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
| 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
−7.82966859075252026924381821420, −6.82281493094582201405816710861, −6.10746047959368033018013707523, −5.58421787923883335116587904984, −4.91178930390267536974734540845, −3.93917731038252620341080958779, −3.47152465339704845387986805981, −2.40792437431925600511066285066, −1.34152656801157694558737517749, 0,
1.34152656801157694558737517749, 2.40792437431925600511066285066, 3.47152465339704845387986805981, 3.93917731038252620341080958779, 4.91178930390267536974734540845, 5.58421787923883335116587904984, 6.10746047959368033018013707523, 6.82281493094582201405816710861, 7.82966859075252026924381821420
