| L(s) = 1 | − 1.45·2-s + 2.41·3-s + 0.117·4-s − 2.25·5-s − 3.50·6-s − 1.27·7-s + 2.73·8-s + 2.80·9-s + 3.28·10-s + 2.81·11-s + 0.282·12-s + 1.19·13-s + 1.84·14-s − 5.43·15-s − 4.22·16-s − 3.27·17-s − 4.08·18-s + 4.87·19-s − 0.264·20-s − 3.06·21-s − 4.10·22-s + 9.08·23-s + 6.60·24-s + 0.0853·25-s − 1.74·26-s − 0.458·27-s − 0.149·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.39·3-s + 0.0586·4-s − 1.00·5-s − 1.43·6-s − 0.480·7-s + 0.968·8-s + 0.936·9-s + 1.03·10-s + 0.850·11-s + 0.0816·12-s + 0.332·13-s + 0.494·14-s − 1.40·15-s − 1.05·16-s − 0.795·17-s − 0.963·18-s + 1.11·19-s − 0.0591·20-s − 0.668·21-s − 0.874·22-s + 1.89·23-s + 1.34·24-s + 0.0170·25-s − 0.342·26-s − 0.0882·27-s − 0.0281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102406321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102406321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 - 1.19T + 13T^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 23 | \( 1 - 9.08T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 + 5.98T + 37T^{2} \) |
| 41 | \( 1 + 0.101T + 41T^{2} \) |
| 43 | \( 1 - 0.656T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 - 5.04T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 6.42T + 61T^{2} \) |
| 67 | \( 1 - 1.69T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 + 9.64T + 73T^{2} \) |
| 79 | \( 1 + 5.55T + 79T^{2} \) |
| 83 | \( 1 - 6.25T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−10.28337190029002535530303763496, −9.407212578747756788838488572971, −8.805268938365506362385183707347, −8.310600149549945089144559326403, −7.41577794896918790490720343717, −6.69960857252530947665157764552, −4.71726727247958300999847133889, −3.76367899357503732763504718450, −2.81008976254879807105378062159, −1.06488908327438773038967481836,
1.06488908327438773038967481836, 2.81008976254879807105378062159, 3.76367899357503732763504718450, 4.71726727247958300999847133889, 6.69960857252530947665157764552, 7.41577794896918790490720343717, 8.310600149549945089144559326403, 8.805268938365506362385183707347, 9.407212578747756788838488572971, 10.28337190029002535530303763496
