| L(s) = 1 | − 5.51·2-s + 1.26·3-s + 22.3·4-s + 19.4·5-s − 6.95·6-s + 10.3·7-s − 79.2·8-s − 25.4·9-s − 107.·10-s − 22.9·11-s + 28.2·12-s + 28.1·13-s − 57.3·14-s + 24.5·15-s + 257.·16-s + 54.8·17-s + 140.·18-s − 54.7·19-s + 435.·20-s + 13.1·21-s + 126.·22-s − 70.2·23-s − 99.9·24-s + 253.·25-s − 154.·26-s − 66.1·27-s + 232.·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 0.242·3-s + 2.79·4-s + 1.73·5-s − 0.473·6-s + 0.561·7-s − 3.50·8-s − 0.940·9-s − 3.38·10-s − 0.629·11-s + 0.679·12-s + 0.599·13-s − 1.09·14-s + 0.422·15-s + 4.02·16-s + 0.782·17-s + 1.83·18-s − 0.661·19-s + 4.86·20-s + 0.136·21-s + 1.22·22-s − 0.636·23-s − 0.850·24-s + 2.02·25-s − 1.16·26-s − 0.471·27-s + 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.310897373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310897373\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 5.51T + 8T^{2} \) |
| 3 | \( 1 - 1.26T + 27T^{2} \) |
| 5 | \( 1 - 19.4T + 125T^{2} \) |
| 7 | \( 1 - 10.3T + 343T^{2} \) |
| 11 | \( 1 + 22.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 70.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 73.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 291.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 199.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 298.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 897.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 891.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 820.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 883.T + 9.12e5T^{2} \) |
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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
−8.941758501053244206412454561433, −8.329106877803784312935964596061, −7.66959626603797368576068770530, −6.64673637950696464621377493079, −5.85579770288584973041399493940, −5.43530197068635063504658059940, −3.28372181800375293463320766293, −2.23828020699132144183622604506, −1.85873657649338625754723695722, −0.69799990916330901891609627121,
0.69799990916330901891609627121, 1.85873657649338625754723695722, 2.23828020699132144183622604506, 3.28372181800375293463320766293, 5.43530197068635063504658059940, 5.85579770288584973041399493940, 6.64673637950696464621377493079, 7.66959626603797368576068770530, 8.329106877803784312935964596061, 8.941758501053244206412454561433
