| L(s) = 1 | − 6i·2-s + 92·4-s − 64i·7-s − 1.32e3i·8-s + 948·11-s + 5.09e3i·13-s − 384·14-s + 3.85e3·16-s − 2.83e4i·17-s + 8.62e3·19-s − 5.68e3i·22-s − 1.52e4i·23-s + 3.05e4·26-s − 5.88e3i·28-s + 3.65e4·29-s + ⋯ |
| L(s) = 1 | − 0.530i·2-s + 0.718·4-s − 0.0705i·7-s − 0.911i·8-s + 0.214·11-s + 0.643i·13-s − 0.0374·14-s + 0.235·16-s − 1.40i·17-s + 0.288·19-s − 0.113i·22-s − 0.262i·23-s + 0.341·26-s − 0.0506i·28-s + 0.277·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.380285569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.380285569\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 6iT - 128T^{2} \) |
| 7 | \( 1 + 64iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 948T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.09e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.83e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 8.62e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.52e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 3.65e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.76e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.68e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.85e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.83e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 4.28e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 1.30e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.00e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.07e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.56e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.36e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 6.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.37e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 8.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.82e6iT - 8.07e13T^{2} \) |
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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
−10.82280912247434167234650656206, −9.814465001916583447003026420218, −8.929103900416934367891360769240, −7.43186669446770798760963195365, −6.79156762869029948438089217545, −5.50126743160538765753684676405, −4.08716755828196382330466598274, −2.90505598717356100377961163292, −1.82632877636914417024830143450, −0.55438036008380261228286944237,
1.29531014608654992542475489573, 2.55003598672261411497073527977, 3.87555893189226680981403347497, 5.46499258685382484333402600151, 6.17850407807383035821864066716, 7.33720912230895446627296708931, 8.091605807919504353675571647858, 9.217794542987190426455789259240, 10.52694459073778741584251315878, 11.15105095658624620173158475341
