| L(s) = 1 | + 0.817·2-s − 0.0745·3-s − 7.33·4-s − 0.0609·6-s − 12.0·7-s − 12.5·8-s − 26.9·9-s − 60.0·11-s + 0.546·12-s + 80.2·13-s − 9.82·14-s + 48.4·16-s − 17.0·17-s − 22.0·18-s − 19·19-s + 0.896·21-s − 49.1·22-s + 149.·23-s + 0.934·24-s + 65.6·26-s + 4.02·27-s + 88.1·28-s + 150.·29-s + 22.1·31-s + 139.·32-s + 4.48·33-s − 13.9·34-s + ⋯ |
| L(s) = 1 | + 0.289·2-s − 0.0143·3-s − 0.916·4-s − 0.00414·6-s − 0.649·7-s − 0.553·8-s − 0.999·9-s − 1.64·11-s + 0.0131·12-s + 1.71·13-s − 0.187·14-s + 0.756·16-s − 0.243·17-s − 0.289·18-s − 0.229·19-s + 0.00931·21-s − 0.476·22-s + 1.35·23-s + 0.00794·24-s + 0.495·26-s + 0.0286·27-s + 0.594·28-s + 0.961·29-s + 0.128·31-s + 0.772·32-s + 0.0236·33-s − 0.0704·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.089983376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089983376\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 0.817T + 8T^{2} \) |
| 3 | \( 1 + 0.0745T + 27T^{2} \) |
| 7 | \( 1 + 12.0T + 343T^{2} \) |
| 11 | \( 1 + 60.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 22.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 68.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 35.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 371.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 655.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 389.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 450.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.21e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 792.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 494.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 205.T + 9.12e5T^{2} \) |
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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
−10.62331697142489242294471293003, −9.691420022813571423386145238082, −8.495071987756110651577678568628, −8.369392687691221036948659263372, −6.70730073087486580030942992696, −5.70223074489771181994032095260, −4.99720523222875305583446886469, −3.62229228105135447277089150531, −2.79098792930315900920969880407, −0.61380919104787783641752969771,
0.61380919104787783641752969771, 2.79098792930315900920969880407, 3.62229228105135447277089150531, 4.99720523222875305583446886469, 5.70223074489771181994032095260, 6.70730073087486580030942992696, 8.369392687691221036948659263372, 8.495071987756110651577678568628, 9.691420022813571423386145238082, 10.62331697142489242294471293003
