| L(s) = 1 | − 2-s − 2.91·3-s + 4-s + 2.71·5-s + 2.91·6-s + 1.04·7-s − 8-s + 5.48·9-s − 2.71·10-s + 1.91·11-s − 2.91·12-s − 0.807·13-s − 1.04·14-s − 7.90·15-s + 16-s + 3.06·17-s − 5.48·18-s + 19-s + 2.71·20-s − 3.04·21-s − 1.91·22-s + 8.03·23-s + 2.91·24-s + 2.36·25-s + 0.807·26-s − 7.23·27-s + 1.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.68·3-s + 0.5·4-s + 1.21·5-s + 1.18·6-s + 0.395·7-s − 0.353·8-s + 1.82·9-s − 0.858·10-s + 0.577·11-s − 0.840·12-s − 0.223·13-s − 0.279·14-s − 2.04·15-s + 0.250·16-s + 0.744·17-s − 1.29·18-s + 0.229·19-s + 0.606·20-s − 0.664·21-s − 0.408·22-s + 1.67·23-s + 0.594·24-s + 0.473·25-s + 0.158·26-s − 1.39·27-s + 0.197·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265380590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265380590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.91T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 0.807T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 23 | \( 1 - 8.03T + 23T^{2} \) |
| 29 | \( 1 - 3.67T + 29T^{2} \) |
| 31 | \( 1 + 5.42T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 3.81T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 + 1.08T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 6.89T + 83T^{2} \) |
| 89 | \( 1 - 1.63T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.51805371780101009652748247448, −7.07767553390248706620130155755, −6.41632508070296403711296220744, −5.72683400588499035511837734658, −5.34044935135831752413444742815, −4.66543341381635875852913611303, −3.49518151750760369735044223533, −2.28003190691578012536246755548, −1.38555945455178741362317308922, −0.76648161673780212670720308369,
0.76648161673780212670720308369, 1.38555945455178741362317308922, 2.28003190691578012536246755548, 3.49518151750760369735044223533, 4.66543341381635875852913611303, 5.34044935135831752413444742815, 5.72683400588499035511837734658, 6.41632508070296403711296220744, 7.07767553390248706620130155755, 7.51805371780101009652748247448
