| L(s) = 1 | − 2.39·2-s + 3.73·4-s + 1.73·5-s − 1.75·7-s − 4.14·8-s − 4.14·10-s + 2·11-s + 2.46·13-s + 4.19·14-s + 2.46·16-s + 3.19·19-s + 6.46·20-s − 4.78·22-s − 1.53·23-s − 2.00·25-s − 5.89·26-s − 6.54·28-s − 1.73·29-s − 9.57·31-s + 2.39·32-s − 3.03·35-s + 10.0·37-s − 7.65·38-s − 7.18·40-s + 10.6·41-s − 3.73·43-s + 7.46·44-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.86·4-s + 0.774·5-s − 0.662·7-s − 1.46·8-s − 1.31·10-s + 0.603·11-s + 0.683·13-s + 1.12·14-s + 0.616·16-s + 0.733·19-s + 1.44·20-s − 1.02·22-s − 0.320·23-s − 0.400·25-s − 1.15·26-s − 1.23·28-s − 0.321·29-s − 1.72·31-s + 0.423·32-s − 0.513·35-s + 1.65·37-s − 1.24·38-s − 1.13·40-s + 1.66·41-s − 0.569·43-s + 1.12·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9477837297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9477837297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 9.57T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 3.03T + 73T^{2} \) |
| 79 | \( 1 + 7.82T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
| 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
−7.916307753973703214515269147841, −7.36679875510094938322333688285, −6.56374776777561661723739201010, −6.11114355476955943336552791159, −5.41248176701585342978368169148, −4.11175821591785771036656512434, −3.26601454483731431835599146032, −2.27248337529066737729467379761, −1.56617192724139837860602501569, −0.65004160257863599080926783591,
0.65004160257863599080926783591, 1.56617192724139837860602501569, 2.27248337529066737729467379761, 3.26601454483731431835599146032, 4.11175821591785771036656512434, 5.41248176701585342978368169148, 6.11114355476955943336552791159, 6.56374776777561661723739201010, 7.36679875510094938322333688285, 7.916307753973703214515269147841
