| L(s) = 1 | − 2-s − 1.64·3-s + 4-s − 4.02·5-s + 1.64·6-s + 2.75·7-s − 8-s − 0.297·9-s + 4.02·10-s − 1.24·11-s − 1.64·12-s − 0.823·13-s − 2.75·14-s + 6.62·15-s + 16-s − 6.05·17-s + 0.297·18-s + 0.349·19-s − 4.02·20-s − 4.52·21-s + 1.24·22-s − 2.04·23-s + 1.64·24-s + 11.2·25-s + 0.823·26-s + 5.42·27-s + 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.949·3-s + 0.5·4-s − 1.80·5-s + 0.671·6-s + 1.04·7-s − 0.353·8-s − 0.0990·9-s + 1.27·10-s − 0.376·11-s − 0.474·12-s − 0.228·13-s − 0.736·14-s + 1.71·15-s + 0.250·16-s − 1.46·17-s + 0.0700·18-s + 0.0802·19-s − 0.900·20-s − 0.988·21-s + 0.266·22-s − 0.426·23-s + 0.335·24-s + 2.24·25-s + 0.161·26-s + 1.04·27-s + 0.520·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1377887767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1377887767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 13 | \( 1 + 0.823T + 13T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 19 | \( 1 - 0.349T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 + 0.869T + 29T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 8.19T + 41T^{2} \) |
| 43 | \( 1 + 4.72T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 7.47T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 3.84T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 1.71T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
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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
−8.075773402582964582863084459627, −7.50305730541383188790899908928, −6.83982473506661493192780362498, −6.09958009041305564473270937619, −4.96644683236255795492453979788, −4.68917096932243980773798240220, −3.72357369282556708061844593092, −2.72203972077098874286600033200, −1.52936012022931725860134253889, −0.23078624103897618521146060832,
0.23078624103897618521146060832, 1.52936012022931725860134253889, 2.72203972077098874286600033200, 3.72357369282556708061844593092, 4.68917096932243980773798240220, 4.96644683236255795492453979788, 6.09958009041305564473270937619, 6.83982473506661493192780362498, 7.50305730541383188790899908928, 8.075773402582964582863084459627
