| L(s) = 1 | + 52.7·3-s − 1.90e3i·5-s + (2.47e3 + 5.84e3i)7-s − 1.68e4·9-s − 6.33e4i·11-s + 1.86e5i·13-s − 1.00e5i·15-s + 2.16e5i·17-s − 1.84e5·19-s + (1.30e5 + 3.08e5i)21-s + 1.40e6i·23-s − 1.66e6·25-s − 1.93e6·27-s − 3.61e4·29-s + 4.89e6·31-s + ⋯ |
| L(s) = 1 | + 0.376·3-s − 1.36i·5-s + (0.389 + 0.920i)7-s − 0.858·9-s − 1.30i·11-s + 1.80i·13-s − 0.512i·15-s + 0.629i·17-s − 0.325·19-s + (0.146 + 0.346i)21-s + 1.04i·23-s − 0.853·25-s − 0.699·27-s − 0.00948·29-s + 0.952·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.838008404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.838008404\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.47e3 - 5.84e3i)T \) |
| good | 3 | \( 1 - 52.7T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.90e3iT - 1.95e6T^{2} \) |
| 11 | \( 1 + 6.33e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.86e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 2.16e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 1.84e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.40e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 3.61e4T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.89e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.90e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.47e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 6.75e6iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 3.59e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.50e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.05e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.47e6iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 3.11e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 4.58e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 - 1.23e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 2.43e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 5.66e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.22e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 - 1.51e9iT - 7.60e17T^{2} \) |
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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
−11.81128859096902938737030846112, −11.37235761844110731455220483897, −9.407448473486793044705845474036, −8.719300638097530880358005571127, −8.182810900295899497313183092429, −6.19234701167568880839060058776, −5.25960787411526872942462584360, −3.97445806578680962449546632973, −2.36801450322252349349970720869, −1.12087105986513250840448678538,
0.48331081193657706564051907509, 2.39417858920620940791953738082, 3.24048293638000004402061236135, 4.73415192890849648652746170316, 6.30772094771624628001032817165, 7.41134498441934413481983737644, 8.159359210708308330090529680750, 9.883136952049959596197158562129, 10.54887032666953688872522681618, 11.46410848021769827359284462590
