| L(s) = 1 | + 1.99·2-s + 1.98·4-s − 2.38·5-s − 0.101·7-s − 0.0328·8-s − 4.76·10-s + 3.79·11-s − 3.75·13-s − 0.201·14-s − 4.03·16-s + 4.24·17-s − 3.15·19-s − 4.73·20-s + 7.56·22-s + 23-s + 0.689·25-s − 7.49·26-s − 0.200·28-s + 29-s + 0.905·31-s − 7.98·32-s + 8.48·34-s + 0.241·35-s + 6.40·37-s − 6.29·38-s + 0.0784·40-s + 3.03·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.991·4-s − 1.06·5-s − 0.0382·7-s − 0.0116·8-s − 1.50·10-s + 1.14·11-s − 1.04·13-s − 0.0539·14-s − 1.00·16-s + 1.03·17-s − 0.723·19-s − 1.05·20-s + 1.61·22-s + 0.208·23-s + 0.137·25-s − 1.46·26-s − 0.0378·28-s + 0.185·29-s + 0.162·31-s − 1.41·32-s + 1.45·34-s + 0.0407·35-s + 1.05·37-s − 1.02·38-s + 0.0124·40-s + 0.473·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.185252728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.185252728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 + 0.101T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 31 | \( 1 - 0.905T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 - 3.03T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 6.65T + 73T^{2} \) |
| 79 | \( 1 - 7.81T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
| 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
−7.75945992391614748540090192081, −7.31856290598094823176042333193, −6.47769188395544220403050903751, −5.86255314383904930022848300953, −5.03493519252625005721729380088, −4.21962230356982039125591759840, −3.96602790702728177742032248950, −3.07893142950744682315198555490, −2.26283924639269270521867918761, −0.74625448732147279613834725990,
0.74625448732147279613834725990, 2.26283924639269270521867918761, 3.07893142950744682315198555490, 3.96602790702728177742032248950, 4.21962230356982039125591759840, 5.03493519252625005721729380088, 5.86255314383904930022848300953, 6.47769188395544220403050903751, 7.31856290598094823176042333193, 7.75945992391614748540090192081
