| L(s) = 1 | − 2.67·2-s + 0.688·3-s + 5.17·4-s − 3.33·5-s − 1.84·6-s + 3.78·7-s − 8.50·8-s − 2.52·9-s + 8.92·10-s + 1.45·11-s + 3.56·12-s + 3.73·13-s − 10.1·14-s − 2.29·15-s + 12.4·16-s − 0.457·17-s + 6.76·18-s − 4.11·19-s − 17.2·20-s + 2.60·21-s − 3.88·22-s − 6.37·23-s − 5.85·24-s + 6.11·25-s − 10.0·26-s − 3.80·27-s + 19.5·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.397·3-s + 2.58·4-s − 1.49·5-s − 0.753·6-s + 1.43·7-s − 3.00·8-s − 0.841·9-s + 2.82·10-s + 0.437·11-s + 1.02·12-s + 1.03·13-s − 2.71·14-s − 0.593·15-s + 3.10·16-s − 0.110·17-s + 1.59·18-s − 0.944·19-s − 3.85·20-s + 0.569·21-s − 0.828·22-s − 1.33·23-s − 1.19·24-s + 1.22·25-s − 1.96·26-s − 0.732·27-s + 3.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 - 0.688T + 3T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 0.457T + 17T^{2} \) |
| 19 | \( 1 + 4.11T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 + 7.35T + 29T^{2} \) |
| 31 | \( 1 - 5.96T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 7.78T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 4.14T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 9.49T + 97T^{2} \) |
| 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
−7.84680072112885242492288069970, −7.37496794518496224882941521919, −6.31796771909310535818961936658, −5.79625191946648906640459573167, −4.36755101149841813579381960430, −3.82514414398002561546205045650, −2.75171749978371224911172875945, −1.94685055431516318860214379946, −1.04215725808419503501793416488, 0,
1.04215725808419503501793416488, 1.94685055431516318860214379946, 2.75171749978371224911172875945, 3.82514414398002561546205045650, 4.36755101149841813579381960430, 5.79625191946648906640459573167, 6.31796771909310535818961936658, 7.37496794518496224882941521919, 7.84680072112885242492288069970
