| L(s) = 1 | + 1.61·2-s − 3-s + 0.618·4-s − 5-s − 1.61·6-s − 1.23·7-s − 2.23·8-s + 9-s − 1.61·10-s − 0.618·12-s + 0.763·13-s − 2.00·14-s + 15-s − 4.85·16-s + 3.47·17-s + 1.61·18-s − 2.47·19-s − 0.618·20-s + 1.23·21-s + 8.70·23-s + 2.23·24-s + 25-s + 1.23·26-s − 27-s − 0.763·28-s + 3.23·29-s + 1.61·30-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.447·5-s − 0.660·6-s − 0.467·7-s − 0.790·8-s + 0.333·9-s − 0.511·10-s − 0.178·12-s + 0.211·13-s − 0.534·14-s + 0.258·15-s − 1.21·16-s + 0.842·17-s + 0.381·18-s − 0.567·19-s − 0.138·20-s + 0.269·21-s + 1.81·23-s + 0.456·24-s + 0.200·25-s + 0.242·26-s − 0.192·27-s − 0.144·28-s + 0.600·29-s + 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944862464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944862464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 0.763T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 - 6.76T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 1.52T + 73T^{2} \) |
| 79 | \( 1 + 0.708T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−9.325012064678932692986700752254, −8.460909499670961194537051937551, −7.47676152222131388133449886600, −6.50230302422962949453235552369, −6.04276253560562332359592948124, −5.00708051626643676653931728210, −4.48527348226146045173814460850, −3.48742788429213568835127407996, −2.74253557029201212677306955913, −0.830149055478123674746407958472,
0.830149055478123674746407958472, 2.74253557029201212677306955913, 3.48742788429213568835127407996, 4.48527348226146045173814460850, 5.00708051626643676653931728210, 6.04276253560562332359592948124, 6.50230302422962949453235552369, 7.47676152222131388133449886600, 8.460909499670961194537051937551, 9.325012064678932692986700752254
