| L(s) = 1 | − 1.20·3-s + 1.10·5-s + 3.71·7-s − 1.55·9-s − 3.29·11-s − 0.161·13-s − 1.32·15-s − 17-s − 7.78·19-s − 4.46·21-s + 2.77·23-s − 3.77·25-s + 5.47·27-s + 4.90·29-s + 2.51·31-s + 3.95·33-s + 4.10·35-s + 7.97·37-s + 0.193·39-s − 2.25·41-s + 3.66·43-s − 1.71·45-s + 11.1·47-s + 6.78·49-s + 1.20·51-s − 3.20·53-s − 3.64·55-s + ⋯ |
| L(s) = 1 | − 0.694·3-s + 0.494·5-s + 1.40·7-s − 0.518·9-s − 0.993·11-s − 0.0447·13-s − 0.343·15-s − 0.242·17-s − 1.78·19-s − 0.973·21-s + 0.578·23-s − 0.755·25-s + 1.05·27-s + 0.910·29-s + 0.451·31-s + 0.689·33-s + 0.693·35-s + 1.31·37-s + 0.0310·39-s − 0.352·41-s + 0.559·43-s − 0.256·45-s + 1.63·47-s + 0.968·49-s + 0.168·51-s − 0.440·53-s − 0.491·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.20T + 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 - 3.71T + 7T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 0.161T + 13T^{2} \) |
| 19 | \( 1 + 7.78T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 - 7.97T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 - 3.66T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 3.20T + 53T^{2} \) |
| 61 | \( 1 + 2.54T + 61T^{2} \) |
| 67 | \( 1 - 0.497T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 4.99T + 83T^{2} \) |
| 89 | \( 1 - 0.726T + 89T^{2} \) |
| 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
−7.52264584706190950951772108875, −6.70366720616659165205008307705, −5.83617995962269125413537023841, −5.59429313692764879924127435844, −4.56591042590417753890505047000, −4.39371125042938861620689787973, −2.79341147494923932986812623105, −2.27429582641824693422986700765, −1.23601436812344903233454829887, 0,
1.23601436812344903233454829887, 2.27429582641824693422986700765, 2.79341147494923932986812623105, 4.39371125042938861620689787973, 4.56591042590417753890505047000, 5.59429313692764879924127435844, 5.83617995962269125413537023841, 6.70366720616659165205008307705, 7.52264584706190950951772108875
