| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 10.9·5-s + 6·6-s + 3.94·7-s + 8·8-s + 9·9-s + 21.8·10-s + 40.1·11-s + 12·12-s + 9.12·13-s + 7.89·14-s + 32.7·15-s + 16·16-s − 99.7·17-s + 18·18-s − 68.7·19-s + 43.7·20-s + 11.8·21-s + 80.3·22-s + 122.·23-s + 24·24-s − 5.47·25-s + 18.2·26-s + 27·27-s + 15.7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.977·5-s + 0.408·6-s + 0.213·7-s + 0.353·8-s + 0.333·9-s + 0.691·10-s + 1.10·11-s + 0.288·12-s + 0.194·13-s + 0.150·14-s + 0.564·15-s + 0.250·16-s − 1.42·17-s + 0.235·18-s − 0.829·19-s + 0.488·20-s + 0.122·21-s + 0.778·22-s + 1.10·23-s + 0.204·24-s − 0.0438·25-s + 0.137·26-s + 0.192·27-s + 0.106·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.473633130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.473633130\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
| good | 5 | \( 1 - 10.9T + 125T^{2} \) |
| 7 | \( 1 - 3.94T + 343T^{2} \) |
| 11 | \( 1 - 40.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.12T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 59.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 406.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 403.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 560.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 131.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 846.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 250.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 413.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 407.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 634.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 455.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 814.T + 9.12e5T^{2} \) |
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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
−11.11330480346808387524662235501, −10.13774373563577144961054940174, −9.134347297434410014993125267158, −8.395703192500064435620901130004, −6.84197878824549284425629040668, −6.33703668533658619851300313942, −4.97364944432932615478524717398, −3.98249948594592295646067540254, −2.60096096387622096766341504944, −1.53714635485635091826750530674,
1.53714635485635091826750530674, 2.60096096387622096766341504944, 3.98249948594592295646067540254, 4.97364944432932615478524717398, 6.33703668533658619851300313942, 6.84197878824549284425629040668, 8.395703192500064435620901130004, 9.134347297434410014993125267158, 10.13774373563577144961054940174, 11.11330480346808387524662235501
