| L(s) = 1 | − 2-s − 0.372·3-s + 4-s + 0.124·5-s + 0.372·6-s − 2.88·7-s − 8-s − 2.86·9-s − 0.124·10-s − 5.02·11-s − 0.372·12-s − 3.71·13-s + 2.88·14-s − 0.0464·15-s + 16-s + 3.34·17-s + 2.86·18-s − 3.40·19-s + 0.124·20-s + 1.07·21-s + 5.02·22-s + 8.69·23-s + 0.372·24-s − 4.98·25-s + 3.71·26-s + 2.18·27-s − 2.88·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.215·3-s + 0.5·4-s + 0.0557·5-s + 0.152·6-s − 1.09·7-s − 0.353·8-s − 0.953·9-s − 0.0394·10-s − 1.51·11-s − 0.107·12-s − 1.03·13-s + 0.770·14-s − 0.0119·15-s + 0.250·16-s + 0.812·17-s + 0.674·18-s − 0.781·19-s + 0.0278·20-s + 0.234·21-s + 1.07·22-s + 1.81·23-s + 0.0760·24-s − 0.996·25-s + 0.728·26-s + 0.420·27-s − 0.545·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2421889989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2421889989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.372T + 3T^{2} \) |
| 5 | \( 1 - 0.124T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 - 8.69T + 23T^{2} \) |
| 29 | \( 1 - 0.896T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 + 4.99T + 47T^{2} \) |
| 53 | \( 1 - 6.24T + 53T^{2} \) |
| 59 | \( 1 + 0.943T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 - 5.92T + 83T^{2} \) |
| 89 | \( 1 - 4.98T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−8.519175295595765255684395885708, −7.69600603858508523467241752206, −7.09813311927891745894224611248, −6.34862212230359499925482562080, −5.41076054792700729182547250160, −5.07681542874794783619785243225, −3.43801183500034661199501384655, −2.91925330065895774182500187969, −2.01720919284463426714356865990, −0.29099780834472831625347116865,
0.29099780834472831625347116865, 2.01720919284463426714356865990, 2.91925330065895774182500187969, 3.43801183500034661199501384655, 5.07681542874794783619785243225, 5.41076054792700729182547250160, 6.34862212230359499925482562080, 7.09813311927891745894224611248, 7.69600603858508523467241752206, 8.519175295595765255684395885708
