| L(s) = 1 | + 5.53·3-s + 15.3·5-s + 23.3·7-s + 3.60·9-s + 16.3·11-s + 8.97·13-s + 84.8·15-s + 124.·17-s + 19·19-s + 129.·21-s + 21.6·23-s + 110.·25-s − 129.·27-s − 123.·29-s − 142.·31-s + 90.5·33-s + 358.·35-s − 177.·37-s + 49.6·39-s + 133.·41-s + 239.·43-s + 55.3·45-s − 129.·47-s + 203.·49-s + 686.·51-s − 172.·53-s + 251.·55-s + ⋯ |
| L(s) = 1 | + 1.06·3-s + 1.37·5-s + 1.26·7-s + 0.133·9-s + 0.448·11-s + 0.191·13-s + 1.46·15-s + 1.77·17-s + 0.229·19-s + 1.34·21-s + 0.196·23-s + 0.880·25-s − 0.922·27-s − 0.788·29-s − 0.823·31-s + 0.477·33-s + 1.73·35-s − 0.790·37-s + 0.203·39-s + 0.507·41-s + 0.848·43-s + 0.183·45-s − 0.400·47-s + 0.592·49-s + 1.88·51-s − 0.446·53-s + 0.615·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.259575336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.259575336\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 5.53T + 27T^{2} \) |
| 5 | \( 1 - 15.3T + 125T^{2} \) |
| 7 | \( 1 - 23.3T + 343T^{2} \) |
| 11 | \( 1 - 16.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.97T + 2.19e3T^{2} \) |
| 17 | \( 1 - 124.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 21.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 142.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 634.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 774.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 960.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.41T + 3.89e5T^{2} \) |
| 79 | \( 1 + 415.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 509.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−9.383350416616415985604001386646, −8.558340386960238652830113895975, −7.905179206408249607094453667943, −7.06595929722502480957958319725, −5.69068500741322555140034419323, −5.38962462912634021041363396557, −3.99829436080285744706769516092, −2.98386692403858892165840218991, −1.94382410686539140202309918724, −1.29978487379015465396833371561,
1.29978487379015465396833371561, 1.94382410686539140202309918724, 2.98386692403858892165840218991, 3.99829436080285744706769516092, 5.38962462912634021041363396557, 5.69068500741322555140034419323, 7.06595929722502480957958319725, 7.905179206408249607094453667943, 8.558340386960238652830113895975, 9.383350416616415985604001386646
