| L(s) = 1 | + 2.35·2-s + 3·3-s − 2.45·4-s − 5·5-s + 7.06·6-s − 3.94·7-s − 24.6·8-s + 9·9-s − 11.7·10-s + 42.8·11-s − 7.36·12-s + 36.4·13-s − 9.28·14-s − 15·15-s − 38.3·16-s + 17.8·17-s + 21.1·18-s + 84.6·19-s + 12.2·20-s − 11.8·21-s + 100.·22-s + 93.9·23-s − 73.8·24-s + 25·25-s + 85.7·26-s + 27·27-s + 9.68·28-s + ⋯ |
| L(s) = 1 | + 0.832·2-s + 0.577·3-s − 0.306·4-s − 0.447·5-s + 0.480·6-s − 0.212·7-s − 1.08·8-s + 0.333·9-s − 0.372·10-s + 1.17·11-s − 0.177·12-s + 0.777·13-s − 0.177·14-s − 0.258·15-s − 0.598·16-s + 0.254·17-s + 0.277·18-s + 1.02·19-s + 0.137·20-s − 0.122·21-s + 0.977·22-s + 0.851·23-s − 0.628·24-s + 0.200·25-s + 0.647·26-s + 0.192·27-s + 0.0653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.077837311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.077837311\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 2.35T + 8T^{2} \) |
| 7 | \( 1 + 3.94T + 343T^{2} \) |
| 11 | \( 1 - 42.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.9T + 1.21e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 61.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 397.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 50.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 476.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 75.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 698.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 44.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 669.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 269.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 869.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 9.85T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.52e3T + 9.12e5T^{2} \) |
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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
−10.89541041851310288381176016567, −9.519660079872534109945597468954, −9.028331449204441690278568412083, −8.045327801225051610963463802897, −6.86991885796750342452935594999, −5.89483163632661521977696981583, −4.66717462332638380868509046146, −3.76312476464863904994612130020, −3.01970481541496632512957154467, −1.04091194272243877728865867311,
1.04091194272243877728865867311, 3.01970481541496632512957154467, 3.76312476464863904994612130020, 4.66717462332638380868509046146, 5.89483163632661521977696981583, 6.86991885796750342452935594999, 8.045327801225051610963463802897, 9.028331449204441690278568412083, 9.519660079872534109945597468954, 10.89541041851310288381176016567
