| L(s) = 1 | − 2-s + 4-s + 0.391·5-s − 1.71·7-s − 8-s − 0.391·10-s + 11-s + 3.71·13-s + 1.71·14-s + 16-s − 5.43·17-s − 19-s + 0.391·20-s − 22-s + 3.43·23-s − 4.84·25-s − 3.71·26-s − 1.71·28-s + 3.82·29-s − 3.06·31-s − 32-s + 5.43·34-s − 0.672·35-s − 4.65·37-s + 38-s − 0.391·40-s − 1.06·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.175·5-s − 0.649·7-s − 0.353·8-s − 0.123·10-s + 0.301·11-s + 1.03·13-s + 0.459·14-s + 0.250·16-s − 1.31·17-s − 0.229·19-s + 0.0875·20-s − 0.213·22-s + 0.716·23-s − 0.969·25-s − 0.729·26-s − 0.324·28-s + 0.710·29-s − 0.550·31-s − 0.176·32-s + 0.932·34-s − 0.113·35-s − 0.765·37-s + 0.162·38-s − 0.0618·40-s − 0.166·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.391T + 5T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 - 1.21T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 + 1.04T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 4.65T + 97T^{2} \) |
| 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
−8.352634935465310717324243718451, −7.40544782079716733687653102695, −6.57748010026919893675054138239, −6.26425868437162378201018781219, −5.25640374435267706145709799033, −4.15265529954075732588312975502, −3.36205253868055070694629869591, −2.35041655757658039750242182638, −1.35540547323113083649514877487, 0,
1.35540547323113083649514877487, 2.35041655757658039750242182638, 3.36205253868055070694629869591, 4.15265529954075732588312975502, 5.25640374435267706145709799033, 6.26425868437162378201018781219, 6.57748010026919893675054138239, 7.40544782079716733687653102695, 8.352634935465310717324243718451
