| L(s) = 1 | − 3-s + 0.413·5-s + 2.75·7-s + 9-s − 5.40·11-s + 2.14·13-s − 0.413·15-s + 5.24·17-s − 3.94·19-s − 2.75·21-s + 23-s − 4.82·25-s − 27-s − 29-s + 4.35·31-s + 5.40·33-s + 1.13·35-s − 2.72·37-s − 2.14·39-s − 6.03·41-s − 3.64·43-s + 0.413·45-s + 2.47·47-s + 0.566·49-s − 5.24·51-s − 7.66·53-s − 2.23·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.184·5-s + 1.03·7-s + 0.333·9-s − 1.63·11-s + 0.596·13-s − 0.106·15-s + 1.27·17-s − 0.904·19-s − 0.600·21-s + 0.208·23-s − 0.965·25-s − 0.192·27-s − 0.185·29-s + 0.781·31-s + 0.941·33-s + 0.192·35-s − 0.447·37-s − 0.344·39-s − 0.942·41-s − 0.555·43-s + 0.0616·45-s + 0.361·47-s + 0.0809·49-s − 0.735·51-s − 1.05·53-s − 0.301·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.413T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 0.981T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 3.02T + 79T^{2} \) |
| 83 | \( 1 + 0.833T + 83T^{2} \) |
| 89 | \( 1 + 5.00T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.65998439827634211438073859869, −6.76167342882776468137107902322, −5.94412554977968986306567355591, −5.34391435129426568098409944105, −4.90408682429261643942267698943, −4.02606574885553380819981914614, −3.07118895983628251771464566684, −2.09500500187687788701893460618, −1.28718439210837633325223689646, 0,
1.28718439210837633325223689646, 2.09500500187687788701893460618, 3.07118895983628251771464566684, 4.02606574885553380819981914614, 4.90408682429261643942267698943, 5.34391435129426568098409944105, 5.94412554977968986306567355591, 6.76167342882776468137107902322, 7.65998439827634211438073859869
