| L(s) = 1 | − 2.12·2-s − 3·3-s − 3.49·4-s + 5·5-s + 6.36·6-s − 18.5·7-s + 24.3·8-s + 9·9-s − 10.6·10-s + 10.4·12-s − 37.4·13-s + 39.4·14-s − 15·15-s − 23.8·16-s − 130.·17-s − 19.1·18-s − 126.·19-s − 17.4·20-s + 55.7·21-s − 72.3·23-s − 73.1·24-s + 25·25-s + 79.5·26-s − 27·27-s + 64.9·28-s + 305.·29-s + 31.8·30-s + ⋯ |
| L(s) = 1 | − 0.750·2-s − 0.577·3-s − 0.436·4-s + 0.447·5-s + 0.433·6-s − 1.00·7-s + 1.07·8-s + 0.333·9-s − 0.335·10-s + 0.252·12-s − 0.799·13-s + 0.753·14-s − 0.258·15-s − 0.372·16-s − 1.86·17-s − 0.250·18-s − 1.52·19-s − 0.195·20-s + 0.579·21-s − 0.655·23-s − 0.622·24-s + 0.200·25-s + 0.599·26-s − 0.192·27-s + 0.438·28-s + 1.95·29-s + 0.193·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06167761948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06167761948\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.12T + 8T^{2} \) |
| 7 | \( 1 + 18.5T + 343T^{2} \) |
| 13 | \( 1 + 37.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 305.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 45.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 563.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 158.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 582.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 8.99T + 2.26e5T^{2} \) |
| 67 | \( 1 + 544.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 458.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 708.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 413.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 323.T + 9.12e5T^{2} \) |
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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
−8.968993399940401763230432012440, −8.404132064957174095809687088042, −7.21502798151293654962268478032, −6.60266335274460471284644903128, −5.87389717426292352327853899167, −4.66513646196051366916957423540, −4.22365437890944263107251307145, −2.69258385929998755956476723399, −1.66323537697633697771910555314, −0.13264871776040772075484421479,
0.13264871776040772075484421479, 1.66323537697633697771910555314, 2.69258385929998755956476723399, 4.22365437890944263107251307145, 4.66513646196051366916957423540, 5.87389717426292352327853899167, 6.60266335274460471284644903128, 7.21502798151293654962268478032, 8.404132064957174095809687088042, 8.968993399940401763230432012440
