| L(s) = 1 | + 1.57·2-s + 2.87·3-s + 0.494·4-s + 4.53·6-s + 1.42·7-s − 2.37·8-s + 5.24·9-s + 0.0740·11-s + 1.42·12-s − 5.70·13-s + 2.24·14-s − 4.74·16-s − 17-s + 8.29·18-s − 4.90·19-s + 4.08·21-s + 0.116·22-s + 3.88·23-s − 6.82·24-s − 9.00·26-s + 6.46·27-s + 0.702·28-s + 5.91·29-s + 0.388·31-s − 2.73·32-s + 0.212·33-s − 1.57·34-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 1.65·3-s + 0.247·4-s + 1.85·6-s + 0.536·7-s − 0.840·8-s + 1.74·9-s + 0.0223·11-s + 0.410·12-s − 1.58·13-s + 0.599·14-s − 1.18·16-s − 0.242·17-s + 1.95·18-s − 1.12·19-s + 0.890·21-s + 0.0249·22-s + 0.809·23-s − 1.39·24-s − 1.76·26-s + 1.24·27-s + 0.132·28-s + 1.09·29-s + 0.0697·31-s − 0.484·32-s + 0.0370·33-s − 0.270·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.479775713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.479775713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 3 | \( 1 - 2.87T + 3T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 - 0.0740T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 - 0.388T + 31T^{2} \) |
| 37 | \( 1 - 9.91T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 6.94T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 + 0.0216T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 1.05T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 1.99T + 89T^{2} \) |
| 97 | \( 1 - 5.34T + 97T^{2} \) |
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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
−11.42004263629784203005612258425, −10.07465710087245930010046502586, −9.229188033737928125705920451093, −8.458307703926520166946785752926, −7.57516510337933116993685058450, −6.48130414230619708124518394423, −4.90332292582932388158167201044, −4.31794555354980880037923099556, −3.05995027922568997721690164060, −2.26438068374847289591696769148,
2.26438068374847289591696769148, 3.05995027922568997721690164060, 4.31794555354980880037923099556, 4.90332292582932388158167201044, 6.48130414230619708124518394423, 7.57516510337933116993685058450, 8.458307703926520166946785752926, 9.229188033737928125705920451093, 10.07465710087245930010046502586, 11.42004263629784203005612258425
