| L(s) = 1 | + 1.16·2-s − 0.653·4-s − 2.03·5-s − 0.829·7-s − 3.07·8-s − 2.36·10-s + 3.14·11-s + 5.93·13-s − 0.962·14-s − 2.26·16-s − 0.890·19-s + 1.33·20-s + 3.64·22-s − 1.52·23-s − 0.844·25-s + 6.89·26-s + 0.541·28-s − 5.21·29-s + 2.32·31-s + 3.52·32-s + 1.69·35-s − 6.89·37-s − 1.03·38-s + 6.27·40-s + 3.86·41-s − 8.42·43-s − 2.05·44-s + ⋯ |
| L(s) = 1 | + 0.820·2-s − 0.326·4-s − 0.911·5-s − 0.313·7-s − 1.08·8-s − 0.748·10-s + 0.948·11-s + 1.64·13-s − 0.257·14-s − 0.566·16-s − 0.204·19-s + 0.297·20-s + 0.777·22-s − 0.317·23-s − 0.168·25-s + 1.35·26-s + 0.102·28-s − 0.968·29-s + 0.417·31-s + 0.623·32-s + 0.285·35-s − 1.13·37-s − 0.167·38-s + 0.992·40-s + 0.603·41-s − 1.28·43-s − 0.309·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.817623886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.817623886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 0.829T + 7T^{2} \) |
| 11 | \( 1 - 3.14T + 11T^{2} \) |
| 13 | \( 1 - 5.93T + 13T^{2} \) |
| 19 | \( 1 + 0.890T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 8.42T + 43T^{2} \) |
| 47 | \( 1 - 6.88T + 47T^{2} \) |
| 53 | \( 1 - 8.30T + 53T^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 2.19T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 - 7.73T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.917613527851395310469634845189, −7.00811362803303674830005916824, −6.24876705735129105695602214737, −5.86198466785401937943460057955, −4.88797981476877626801319593465, −4.10912944809265014611618883616, −3.67136083578994914572664934092, −3.22130413033005050966225329895, −1.80193815228895716872661962057, −0.59824258185447156172206348048,
0.59824258185447156172206348048, 1.80193815228895716872661962057, 3.22130413033005050966225329895, 3.67136083578994914572664934092, 4.10912944809265014611618883616, 4.88797981476877626801319593465, 5.86198466785401937943460057955, 6.24876705735129105695602214737, 7.00811362803303674830005916824, 7.917613527851395310469634845189
