| L(s) = 1 | + 1.49·2-s + 1.13·3-s + 1.24·4-s − 5-s + 1.70·6-s + 1.94·7-s + 0.360·8-s + 0.290·9-s − 1.49·10-s − 1.77·11-s + 1.41·12-s + 13-s + 2.90·14-s − 1.13·15-s − 0.700·16-s + 0.241·17-s + 0.435·18-s − 1.24·20-s + 2.20·21-s − 2.65·22-s − 0.709·23-s + 0.410·24-s + 25-s + 1.49·26-s − 0.805·27-s + 2.41·28-s − 1.70·30-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.13·3-s + 1.24·4-s − 5-s + 1.70·6-s + 1.94·7-s + 0.360·8-s + 0.290·9-s − 1.49·10-s − 1.77·11-s + 1.41·12-s + 13-s + 2.90·14-s − 1.13·15-s − 0.700·16-s + 0.241·17-s + 0.435·18-s − 1.24·20-s + 2.20·21-s − 2.65·22-s − 0.709·23-s + 0.410·24-s + 25-s + 1.49·26-s − 0.805·27-s + 2.41·28-s − 1.70·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.289374244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.289374244\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.49T + T^{2} \) |
| 3 | \( 1 - 1.13T + T^{2} \) |
| 7 | \( 1 - 1.94T + T^{2} \) |
| 11 | \( 1 + 1.77T + T^{2} \) |
| 17 | \( 1 - 0.241T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.709T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.49T + T^{2} \) |
| 47 | \( 1 - 0.709T + T^{2} \) |
| 53 | \( 1 - 1.77T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.77T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.94T + T^{2} \) |
| 97 | \( 1 + 1.13T + T^{2} \) |
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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
−8.890492332827095581215994888062, −8.338446593285847186221069951837, −7.83586601130032949656328374155, −7.22297455286393067231523623202, −5.71225615584266994112661659198, −5.16384507324765037444472653402, −4.33237265664693435258092036523, −3.66434342184411936467986469188, −2.79131823306619987267679171894, −1.90786869900449721619756199059,
1.90786869900449721619756199059, 2.79131823306619987267679171894, 3.66434342184411936467986469188, 4.33237265664693435258092036523, 5.16384507324765037444472653402, 5.71225615584266994112661659198, 7.22297455286393067231523623202, 7.83586601130032949656328374155, 8.338446593285847186221069951837, 8.890492332827095581215994888062
