| L(s) = 1 | + 3.17·3-s − 5-s + 4.50·7-s + 7.06·9-s + 5.98·11-s − 2.78·13-s − 3.17·15-s − 1.74·17-s − 2.82·19-s + 14.2·21-s − 2.66·23-s + 25-s + 12.8·27-s − 1.04·29-s − 5.17·31-s + 18.9·33-s − 4.50·35-s + 10.6·37-s − 8.84·39-s − 6.52·41-s − 4.61·43-s − 7.06·45-s + 11.7·47-s + 13.2·49-s − 5.53·51-s + 12.8·53-s − 5.98·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 0.447·5-s + 1.70·7-s + 2.35·9-s + 1.80·11-s − 0.773·13-s − 0.819·15-s − 0.423·17-s − 0.647·19-s + 3.11·21-s − 0.555·23-s + 0.200·25-s + 2.48·27-s − 0.194·29-s − 0.929·31-s + 3.30·33-s − 0.761·35-s + 1.74·37-s − 1.41·39-s − 1.01·41-s − 0.704·43-s − 1.05·45-s + 1.71·47-s + 1.89·49-s − 0.775·51-s + 1.76·53-s − 0.807·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.447836968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.447836968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 - 5.98T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + 1.04T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 6.52T + 41T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
| 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
−8.002132634514895382439311850421, −7.23817025018205215646645951962, −6.94471088248578815244232262691, −5.70078514501130693873770096155, −4.53131791014809619318603749622, −4.22272745024961115454296856886, −3.65993730678663375539121362937, −2.50138070970886570677668779829, −1.93233097827696029638815762520, −1.18190011439970217857635166721,
1.18190011439970217857635166721, 1.93233097827696029638815762520, 2.50138070970886570677668779829, 3.65993730678663375539121362937, 4.22272745024961115454296856886, 4.53131791014809619318603749622, 5.70078514501130693873770096155, 6.94471088248578815244232262691, 7.23817025018205215646645951962, 8.002132634514895382439311850421
