| L(s) = 1 | − 1.29·2-s − 0.310·4-s + 3.52·5-s + 3.00·8-s − 4.58·10-s − 1.17·11-s + 3.22·13-s − 3.28·16-s − 4.90·17-s + 6.86·19-s − 1.09·20-s + 1.53·22-s + 4.29·23-s + 7.43·25-s − 4.18·26-s + 2.72·29-s + 1.92·31-s − 1.73·32-s + 6.37·34-s − 9.76·37-s − 8.92·38-s + 10.5·40-s + 6.65·41-s − 9.66·43-s + 0.365·44-s − 5.58·46-s + 0.633·47-s + ⋯ |
| L(s) = 1 | − 0.919·2-s − 0.155·4-s + 1.57·5-s + 1.06·8-s − 1.44·10-s − 0.355·11-s + 0.893·13-s − 0.820·16-s − 1.18·17-s + 1.57·19-s − 0.244·20-s + 0.326·22-s + 0.896·23-s + 1.48·25-s − 0.821·26-s + 0.505·29-s + 0.344·31-s − 0.307·32-s + 1.09·34-s − 1.60·37-s − 1.44·38-s + 1.67·40-s + 1.03·41-s − 1.47·43-s + 0.0551·44-s − 0.824·46-s + 0.0923·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551445584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551445584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 11 | \( 1 + 1.17T + 11T^{2} \) |
| 13 | \( 1 - 3.22T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 - 6.86T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 + 9.76T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 + 9.66T + 43T^{2} \) |
| 47 | \( 1 - 0.633T + 47T^{2} \) |
| 53 | \( 1 - 2.22T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 + 1.03T + 73T^{2} \) |
| 79 | \( 1 - 1.00T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.804979122486318039579702701675, −7.922234590426718855646783927403, −6.95201305240995737389529712090, −6.44554793651700859396095865824, −5.31251373636192052226630776148, −5.06233341064789630829778323900, −3.78980472270513499137783968998, −2.65040210013164488795150034392, −1.71906183228218707591850954697, −0.883438024388414214238160398805,
0.883438024388414214238160398805, 1.71906183228218707591850954697, 2.65040210013164488795150034392, 3.78980472270513499137783968998, 5.06233341064789630829778323900, 5.31251373636192052226630776148, 6.44554793651700859396095865824, 6.95201305240995737389529712090, 7.922234590426718855646783927403, 8.804979122486318039579702701675
