| L(s) = 1 | − 8·5-s + 9·7-s + 58·11-s + 67·13-s + 38·17-s − 49·19-s − 26·23-s − 61·25-s + 54·29-s + 263·31-s − 72·35-s − 51·37-s + 456·41-s + 199·43-s + 78·47-s − 262·49-s + 442·53-s − 464·55-s − 308·59-s − 914·61-s − 536·65-s − 76·67-s − 430·71-s + 322·73-s + 522·77-s − 875·79-s − 994·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s + 0.485·7-s + 1.58·11-s + 1.42·13-s + 0.542·17-s − 0.591·19-s − 0.235·23-s − 0.487·25-s + 0.345·29-s + 1.52·31-s − 0.347·35-s − 0.226·37-s + 1.73·41-s + 0.705·43-s + 0.242·47-s − 0.763·49-s + 1.14·53-s − 1.13·55-s − 0.679·59-s − 1.91·61-s − 1.02·65-s − 0.138·67-s − 0.718·71-s + 0.516·73-s + 0.772·77-s − 1.24·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.605437199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.605437199\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 58 T + p^{3} T^{2} \) |
| 13 | \( 1 - 67 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 49 T + p^{3} T^{2} \) |
| 23 | \( 1 + 26 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 263 T + p^{3} T^{2} \) |
| 37 | \( 1 + 51 T + p^{3} T^{2} \) |
| 41 | \( 1 - 456 T + p^{3} T^{2} \) |
| 43 | \( 1 - 199 T + p^{3} T^{2} \) |
| 47 | \( 1 - 78 T + p^{3} T^{2} \) |
| 53 | \( 1 - 442 T + p^{3} T^{2} \) |
| 59 | \( 1 + 308 T + p^{3} T^{2} \) |
| 61 | \( 1 + 914 T + p^{3} T^{2} \) |
| 67 | \( 1 + 76 T + p^{3} T^{2} \) |
| 71 | \( 1 + 430 T + p^{3} T^{2} \) |
| 73 | \( 1 - 322 T + p^{3} T^{2} \) |
| 79 | \( 1 + 875 T + p^{3} T^{2} \) |
| 83 | \( 1 + 994 T + p^{3} T^{2} \) |
| 89 | \( 1 + 84 T + p^{3} T^{2} \) |
| 97 | \( 1 - 809 T + p^{3} T^{2} \) |
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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
−8.721842095935323149958063110570, −8.142942111844721534145427235461, −7.34951107920354680866314883944, −6.33892844528396411619369554807, −5.87667494549043539236225481604, −4.38812781468231402033011745056, −4.07584480696521318089101039064, −3.06920921061824305853614575348, −1.61726712255886894733947206841, −0.816430325218280186626035287475,
0.816430325218280186626035287475, 1.61726712255886894733947206841, 3.06920921061824305853614575348, 4.07584480696521318089101039064, 4.38812781468231402033011745056, 5.87667494549043539236225481604, 6.33892844528396411619369554807, 7.34951107920354680866314883944, 8.142942111844721534145427235461, 8.721842095935323149958063110570
