| L(s) = 1 | + 1.13·2-s + 2.38·3-s − 0.707·4-s + 2.71·6-s + 7-s − 3.07·8-s + 2.68·9-s + 3.27·11-s − 1.68·12-s + 4.54·13-s + 1.13·14-s − 2.08·16-s + 7.14·17-s + 3.05·18-s − 8.23·19-s + 2.38·21-s + 3.72·22-s + 1.40·23-s − 7.33·24-s + 5.17·26-s − 0.748·27-s − 0.707·28-s + 4.44·29-s − 2.94·31-s + 3.78·32-s + 7.81·33-s + 8.12·34-s + ⋯ |
| L(s) = 1 | + 0.803·2-s + 1.37·3-s − 0.353·4-s + 1.10·6-s + 0.377·7-s − 1.08·8-s + 0.895·9-s + 0.987·11-s − 0.486·12-s + 1.26·13-s + 0.303·14-s − 0.521·16-s + 1.73·17-s + 0.719·18-s − 1.88·19-s + 0.520·21-s + 0.794·22-s + 0.291·23-s − 1.49·24-s + 1.01·26-s − 0.144·27-s − 0.133·28-s + 0.824·29-s − 0.529·31-s + 0.669·32-s + 1.35·33-s + 1.39·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.414580944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.414580944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 2.38T + 3T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 - 7.14T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 + 5.61T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 + 9.26T + 47T^{2} \) |
| 53 | \( 1 - 0.742T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 4.06T + 61T^{2} \) |
| 67 | \( 1 + 1.20T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 0.957T + 73T^{2} \) |
| 79 | \( 1 + 0.494T + 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 - 9.53T + 89T^{2} \) |
| 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
−9.959332686234166454439050590014, −9.009427603187131127184626067745, −8.596901933097885203016948067303, −7.88094965361732052849348814103, −6.55932952973186408338750693172, −5.72545041543630846605690996840, −4.45241778339775770356995069040, −3.73080592504685339112771051139, −3.02827595954610852028343193959, −1.54446078184689439442197215985,
1.54446078184689439442197215985, 3.02827595954610852028343193959, 3.73080592504685339112771051139, 4.45241778339775770356995069040, 5.72545041543630846605690996840, 6.55932952973186408338750693172, 7.88094965361732052849348814103, 8.596901933097885203016948067303, 9.009427603187131127184626067745, 9.959332686234166454439050590014
