| L(s) = 1 | − 1.23·5-s + 7-s + 2·11-s − 4.47·13-s + 1.23·17-s − 19-s + 2·23-s − 3.47·25-s + 7.23·29-s + 4·31-s − 1.23·35-s + 4.47·37-s − 6.47·41-s − 10.4·43-s + 9.23·47-s + 49-s − 12.1·53-s − 2.47·55-s + 12.9·59-s + 0.472·61-s + 5.52·65-s + 4.94·67-s + 7.23·71-s + 6·73-s + 2·77-s − 16.9·79-s + 6.76·83-s + ⋯ |
| L(s) = 1 | − 0.552·5-s + 0.377·7-s + 0.603·11-s − 1.24·13-s + 0.299·17-s − 0.229·19-s + 0.417·23-s − 0.694·25-s + 1.34·29-s + 0.718·31-s − 0.208·35-s + 0.735·37-s − 1.01·41-s − 1.59·43-s + 1.34·47-s + 0.142·49-s − 1.67·53-s − 0.333·55-s + 1.68·59-s + 0.0604·61-s + 0.685·65-s + 0.604·67-s + 0.858·71-s + 0.702·73-s + 0.227·77-s − 1.90·79-s + 0.742·83-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
| \(L(1)\) |
\(\approx\) |
\(1.635792368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635792368\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 0.472T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.230543230308003743596717645668, −7.61685736823612803634141047484, −6.88209087899201275699280465783, −6.23821385509782428254031713322, −5.14304323514114182231205580808, −4.64246187342294396653448507157, −3.80244999295771040082931127484, −2.89864604317108997233748518703, −1.93521439172926010742921922986, −0.70442958664710639496682572003,
0.70442958664710639496682572003, 1.93521439172926010742921922986, 2.89864604317108997233748518703, 3.80244999295771040082931127484, 4.64246187342294396653448507157, 5.14304323514114182231205580808, 6.23821385509782428254031713322, 6.88209087899201275699280465783, 7.61685736823612803634141047484, 8.230543230308003743596717645668
