| L(s) = 1 | − 128i·2-s − 1.63e4·4-s + (−1.73e5 + 2.01e4i)5-s + 2.89e6i·7-s + 2.09e6i·8-s + (2.57e6 + 2.22e7i)10-s − 6.57e7·11-s − 1.16e8i·13-s + 3.70e8·14-s + 2.68e8·16-s − 1.76e9i·17-s + 7.05e9·19-s + (2.84e9 − 3.29e8i)20-s + 8.41e9i·22-s − 3.96e9i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.993 + 0.115i)5-s + 1.32i·7-s + 0.353i·8-s + (0.0814 + 0.702i)10-s − 1.01·11-s − 0.515i·13-s + 0.938·14-s + 0.250·16-s − 1.04i·17-s + 1.80·19-s + (0.496 − 0.0576i)20-s + 0.719i·22-s − 0.242i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.6257667993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6257667993\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 128iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.73e5 - 2.01e4i)T \) |
| good | 7 | \( 1 - 2.89e6iT - 4.74e12T^{2} \) |
| 11 | \( 1 + 6.57e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.16e8iT - 5.11e16T^{2} \) |
| 17 | \( 1 + 1.76e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 - 7.05e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 3.96e9iT - 2.66e20T^{2} \) |
| 29 | \( 1 - 6.66e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 5.24e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 5.76e9iT - 3.33e23T^{2} \) |
| 41 | \( 1 + 5.49e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.67e12iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 6.04e12iT - 1.20e25T^{2} \) |
| 53 | \( 1 - 3.47e12iT - 7.31e25T^{2} \) |
| 59 | \( 1 + 1.41e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.55e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 8.85e13iT - 2.46e27T^{2} \) |
| 71 | \( 1 + 1.15e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 9.32e13iT - 8.90e27T^{2} \) |
| 79 | \( 1 + 1.53e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 2.32e14iT - 6.11e28T^{2} \) |
| 89 | \( 1 - 1.36e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 5.63e14iT - 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.72186797865187479186192139365, −10.54469742864857266666690533324, −9.373936025597015258662256463287, −8.325763899809668243124938547098, −7.38682499262828866350934794876, −5.59536375375823294546600435108, −4.74783482446405317784756317899, −3.12418148224858983072631882657, −2.63308530138078806997090432414, −0.928466697285059510287389186473,
0.17259398078288212022245572495, 1.19299816938082808516187696860, 3.26450769876545088417808681633, 4.20776740011546524017598912379, 5.22647409528126863289058352855, 6.79028192687208901949102903114, 7.60726191764897891695869423018, 8.326182759172884773767709251171, 9.835778658280478795024995437540, 10.81195416009351436404179464205
