| L(s) = 1 | − 3-s + 0.561·5-s − 5.12·7-s + 9-s − 2.56·11-s + 0.561·13-s − 0.561·15-s + 5.68·17-s − 2.56·19-s + 5.12·21-s + 8·23-s − 4.68·25-s − 27-s + 7.12·29-s + 31-s + 2.56·33-s − 2.87·35-s − 8.24·37-s − 0.561·39-s + 4.24·41-s − 9.12·43-s + 0.561·45-s + 3.68·47-s + 19.2·49-s − 5.68·51-s + 2·53-s − 1.43·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.251·5-s − 1.93·7-s + 0.333·9-s − 0.772·11-s + 0.155·13-s − 0.144·15-s + 1.37·17-s − 0.587·19-s + 1.11·21-s + 1.66·23-s − 0.936·25-s − 0.192·27-s + 1.32·29-s + 0.179·31-s + 0.445·33-s − 0.486·35-s − 1.35·37-s − 0.0899·39-s + 0.663·41-s − 1.39·43-s + 0.0837·45-s + 0.537·47-s + 2.74·49-s − 0.796·51-s + 0.274·53-s − 0.193·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 0.561T + 61T^{2} \) |
| 67 | \( 1 - 0.315T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 0.246T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.61161891375155755242982806839, −6.73795468239627310862273914317, −6.46676276205196497830616060079, −5.55787567720275592806197135974, −5.12290732538520414192068649691, −3.91192896503386462672190751571, −3.21238697513750513160143603969, −2.53010557620620818949075771845, −1.07219916575105933274865241081, 0,
1.07219916575105933274865241081, 2.53010557620620818949075771845, 3.21238697513750513160143603969, 3.91192896503386462672190751571, 5.12290732538520414192068649691, 5.55787567720275592806197135974, 6.46676276205196497830616060079, 6.73795468239627310862273914317, 7.61161891375155755242982806839
