| L(s) = 1 | − 2.64·2-s + 3.40·3-s + 4.98·4-s − 0.0755·5-s − 8.99·6-s − 7.88·8-s + 8.58·9-s + 0.199·10-s + 4.67·11-s + 16.9·12-s + 1.89·13-s − 0.257·15-s + 10.8·16-s − 3.39·17-s − 22.6·18-s − 4.98·19-s − 0.376·20-s − 12.3·22-s + 3.66·23-s − 26.8·24-s − 4.99·25-s − 4.99·26-s + 19.0·27-s − 2.94·29-s + 0.679·30-s − 3.88·31-s − 12.9·32-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 1.96·3-s + 2.49·4-s − 0.0337·5-s − 3.67·6-s − 2.78·8-s + 2.86·9-s + 0.0631·10-s + 1.41·11-s + 4.89·12-s + 0.524·13-s − 0.0664·15-s + 2.71·16-s − 0.822·17-s − 5.34·18-s − 1.14·19-s − 0.0842·20-s − 2.63·22-s + 0.764·23-s − 5.47·24-s − 0.998·25-s − 0.979·26-s + 3.66·27-s − 0.547·29-s + 0.124·30-s − 0.697·31-s − 2.28·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.970428824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970428824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 + 0.0755T + 5T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 - 1.89T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 41 | \( 1 - 5.96T + 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 4.13T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 - 8.25T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 79 | \( 1 + 7.22T + 79T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 - 0.942T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.779768352043033823384905754288, −8.125527422715995247009485109887, −7.39626120239182245715216869774, −6.85358533610953447705662812294, −6.15484250042037192169819008113, −4.21847912343521031913759873427, −3.64366646716928879001690853496, −2.50333514128812222830276099240, −1.95241759598459255814452078316, −1.05473515842970423650415852854,
1.05473515842970423650415852854, 1.95241759598459255814452078316, 2.50333514128812222830276099240, 3.64366646716928879001690853496, 4.21847912343521031913759873427, 6.15484250042037192169819008113, 6.85358533610953447705662812294, 7.39626120239182245715216869774, 8.125527422715995247009485109887, 8.779768352043033823384905754288
