| L(s) = 1 | − 0.147·2-s − 2.43·3-s − 1.97·4-s + 4.40·5-s + 0.359·6-s − 2.90·7-s + 0.587·8-s + 2.93·9-s − 0.649·10-s − 2.89·11-s + 4.82·12-s + 13-s + 0.429·14-s − 10.7·15-s + 3.86·16-s + 2.38·17-s − 0.434·18-s + 5.82·19-s − 8.70·20-s + 7.08·21-s + 0.427·22-s + 4.41·23-s − 1.43·24-s + 14.3·25-s − 0.147·26-s + 0.146·27-s + 5.74·28-s + ⋯ |
| L(s) = 1 | − 0.104·2-s − 1.40·3-s − 0.989·4-s + 1.96·5-s + 0.146·6-s − 1.09·7-s + 0.207·8-s + 0.979·9-s − 0.205·10-s − 0.873·11-s + 1.39·12-s + 0.277·13-s + 0.114·14-s − 2.76·15-s + 0.967·16-s + 0.577·17-s − 0.102·18-s + 1.33·19-s − 1.94·20-s + 1.54·21-s + 0.0912·22-s + 0.920·23-s − 0.292·24-s + 2.87·25-s − 0.0289·26-s + 0.0281·27-s + 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8026531254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8026531254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.147T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 - 4.40T + 5T^{2} \) |
| 7 | \( 1 + 2.90T + 7T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 23 | \( 1 - 4.41T + 23T^{2} \) |
| 29 | \( 1 + 1.08T + 29T^{2} \) |
| 37 | \( 1 - 5.14T + 37T^{2} \) |
| 41 | \( 1 + 2.96T + 41T^{2} \) |
| 43 | \( 1 - 5.32T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2.14T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 8.42T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−10.98095707852679112704323737918, −10.17773032878726788757544041877, −9.700332681855889120392324685715, −8.948506608291361911082356641016, −7.25826691700062241315727549657, −6.04960012671262057143778526075, −5.62987928141843941818819188908, −4.89970988167916250099674069889, −3.00415090684955146393678995174, −0.977114755601746040575345640943,
0.977114755601746040575345640943, 3.00415090684955146393678995174, 4.89970988167916250099674069889, 5.62987928141843941818819188908, 6.04960012671262057143778526075, 7.25826691700062241315727549657, 8.948506608291361911082356641016, 9.700332681855889120392324685715, 10.17773032878726788757544041877, 10.98095707852679112704323737918
