| L(s) = 1 | − 1.95·3-s − 18.0·5-s − 22.0·7-s − 23.1·9-s + 11.1·11-s + 0.532·13-s + 35.2·15-s + 56.2·17-s + 26.5·19-s + 42.9·21-s − 66.9·23-s + 201.·25-s + 97.9·27-s + 29·29-s − 85.2·31-s − 21.8·33-s + 397.·35-s + 180.·37-s − 1.03·39-s + 46.4·41-s + 212.·43-s + 418.·45-s − 108.·47-s + 141.·49-s − 109.·51-s + 444.·53-s − 202.·55-s + ⋯ |
| L(s) = 1 | − 0.375·3-s − 1.61·5-s − 1.18·7-s − 0.859·9-s + 0.306·11-s + 0.0113·13-s + 0.606·15-s + 0.802·17-s + 0.320·19-s + 0.446·21-s − 0.606·23-s + 1.61·25-s + 0.698·27-s + 0.185·29-s − 0.494·31-s − 0.115·33-s + 1.92·35-s + 0.800·37-s − 0.00426·39-s + 0.176·41-s + 0.753·43-s + 1.38·45-s − 0.336·47-s + 0.413·49-s − 0.301·51-s + 1.15·53-s − 0.495·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 1.95T + 27T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 7 | \( 1 + 22.0T + 343T^{2} \) |
| 11 | \( 1 - 11.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.532T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.9T + 1.21e4T^{2} \) |
| 31 | \( 1 + 85.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 46.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 212.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 444.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 43.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 33.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 988.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 154.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 631.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 145.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 169.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 29.4T + 9.12e5T^{2} \) |
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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
−8.381475631456024926776574837651, −7.72440001755032850924864466474, −6.93233688162707639798147717065, −6.13049340973424681331284285151, −5.30962818259831222440442411954, −4.13451603774120340906736468936, −3.50972779520580033447092189958, −2.72629218893552971598967717870, −0.822590028748596357402277513252, 0,
0.822590028748596357402277513252, 2.72629218893552971598967717870, 3.50972779520580033447092189958, 4.13451603774120340906736468936, 5.30962818259831222440442411954, 6.13049340973424681331284285151, 6.93233688162707639798147717065, 7.72440001755032850924864466474, 8.381475631456024926776574837651
