| L(s) = 1 | + 2.73·5-s − 7-s − 3.46·11-s − 4·13-s − 4.19·17-s − 19-s + 7.46·23-s + 2.46·25-s + 6.19·29-s + 4.92·31-s − 2.73·35-s − 7.46·37-s + 9.46·41-s − 3.46·43-s − 2.73·47-s + 49-s − 8.73·53-s − 9.46·55-s + 10.9·59-s − 14.3·61-s − 10.9·65-s + 1.46·67-s − 10.1·71-s − 15.8·73-s + 3.46·77-s − 13.4·79-s + 1.26·83-s + ⋯ |
| L(s) = 1 | + 1.22·5-s − 0.377·7-s − 1.04·11-s − 1.10·13-s − 1.01·17-s − 0.229·19-s + 1.55·23-s + 0.492·25-s + 1.15·29-s + 0.885·31-s − 0.461·35-s − 1.22·37-s + 1.47·41-s − 0.528·43-s − 0.398·47-s + 0.142·49-s − 1.19·53-s − 1.27·55-s + 1.42·59-s − 1.84·61-s − 1.35·65-s + 0.178·67-s − 1.21·71-s − 1.85·73-s + 0.394·77-s − 1.51·79-s + 0.139·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 - 4.92T + 31T^{2} \) |
| 37 | \( 1 + 7.46T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 1.46T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.905762761133778948798963994776, −7.04712134720812364061126020026, −6.52335188093155764935448228034, −5.70424029392471525402457468158, −5.00147348262843879862010636695, −4.43207112370119232712953663037, −2.84220825316301115835751353962, −2.64715394159476842232378922552, −1.49900195174847536244542047421, 0,
1.49900195174847536244542047421, 2.64715394159476842232378922552, 2.84220825316301115835751353962, 4.43207112370119232712953663037, 5.00147348262843879862010636695, 5.70424029392471525402457468158, 6.52335188093155764935448228034, 7.04712134720812364061126020026, 7.905762761133778948798963994776
