L(s) = 1 | + (−200. − 301. i)2-s + 1.34e4i·3-s + (−5.07e4 + 1.20e5i)4-s + 1.59e6i·5-s + (4.06e6 − 2.70e6i)6-s − 1.66e7·7-s + (4.66e7 − 8.91e6i)8-s − 5.26e7·9-s + (4.80e8 − 3.19e8i)10-s − 6.79e8i·11-s + (−1.62e9 − 6.84e8i)12-s + 3.13e9i·13-s + (3.32e9 + 5.00e9i)14-s − 2.14e10·15-s + (−1.20e10 − 1.22e10i)16-s − 1.26e10·17-s + ⋯ |
L(s) = 1 | + (−0.553 − 0.832i)2-s + 1.18i·3-s + (−0.387 + 0.921i)4-s + 1.82i·5-s + (0.988 − 0.656i)6-s − 1.08·7-s + (0.982 − 0.187i)8-s − 0.407·9-s + (1.51 − 1.00i)10-s − 0.955i·11-s + (−1.09 − 0.459i)12-s + 1.06i·13-s + (0.602 + 0.906i)14-s − 2.16·15-s + (−0.700 − 0.714i)16-s − 0.440·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(9)\) |
\(\approx\) |
\(0.0533047 - 0.562565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0533047 - 0.562565i\) |
\(L(\frac{19}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (200. + 301. i)T \) |
good | 3 | \( 1 - 1.34e4iT - 1.29e8T^{2} \) |
| 5 | \( 1 - 1.59e6iT - 7.62e11T^{2} \) |
| 7 | \( 1 + 1.66e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 6.79e8iT - 5.05e17T^{2} \) |
| 13 | \( 1 - 3.13e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + 1.26e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 5.87e9iT - 5.48e21T^{2} \) |
| 23 | \( 1 - 5.43e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.82e12iT - 7.25e24T^{2} \) |
| 31 | \( 1 - 8.42e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 6.17e12iT - 4.56e26T^{2} \) |
| 41 | \( 1 + 6.38e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.41e13iT - 5.87e27T^{2} \) |
| 47 | \( 1 + 3.89e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 5.92e14iT - 2.05e29T^{2} \) |
| 59 | \( 1 - 7.11e14iT - 1.27e30T^{2} \) |
| 61 | \( 1 - 9.01e14iT - 2.24e30T^{2} \) |
| 67 | \( 1 + 8.96e14iT - 1.10e31T^{2} \) |
| 71 | \( 1 - 8.49e13T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.01e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 3.99e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.19e15iT - 4.21e32T^{2} \) |
| 89 | \( 1 + 6.74e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.67e15T + 5.95e33T^{2} \) |
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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
−18.64790310223132575316490930469, −16.75720607538679795631736769814, −15.41086428626393880439518429185, −13.70087527187094083329120225517, −11.32477330192434723387097351598, −10.40471563913043203567805861550, −9.284415707041162343667719877504, −6.79851578780200193348641887051, −3.79970428476851254744366383718, −2.80710324285546941622272471045,
0.29246881389725511467881259021, 1.40750216629179864349843802112, 5.08286517710857280005061792811, 6.79829910045218252234204226791, 8.264009619480325492515380801668, 9.625946911014670383882510287767, 12.66791841868302015744680785973, 13.17623118777747015540708229049, 15.54662487880410782314190218653, 16.80730121902065887771577463969