| L(s) = 1 | − 2i·2-s − 4i·3-s + 28·4-s − 8·6-s − 192i·7-s − 120i·8-s + 227·9-s − 148·11-s − 112i·12-s + 286i·13-s − 384·14-s + 656·16-s + 1.67e3i·17-s − 454i·18-s − 1.06e3·19-s + ⋯ |
| L(s) = 1 | − 0.353i·2-s − 0.256i·3-s + 0.875·4-s − 0.0907·6-s − 1.48i·7-s − 0.662i·8-s + 0.934·9-s − 0.368·11-s − 0.224i·12-s + 0.469i·13-s − 0.523·14-s + 0.640·16-s + 1.40i·17-s − 0.330i·18-s − 0.673·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.49292 - 0.922676i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49292 - 0.922676i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 2iT - 32T^{2} \) |
| 3 | \( 1 + 4iT - 243T^{2} \) |
| 7 | \( 1 + 192iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 148T + 1.61e5T^{2} \) |
| 13 | \( 1 - 286iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.67e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.97e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 182iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 9.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.24e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.20e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.38e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.09e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.87e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.67e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.19e5iT - 8.58e9T^{2} \) |
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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
−16.37724182966403560838305782260, −15.18219705092347137037400602925, −13.55829344747161110588184513815, −12.49753140918593493264913014039, −10.92336839720354410904738185506, −10.09645133090791730157708193439, −7.69068248695973419209856909799, −6.61106280948967233506423682881, −3.93880660793991843123609376745, −1.50030300505208297684140977995,
2.50056475037849419211429099790, 5.27113894087282802360114671500, 6.84299307896800746882379617883, 8.488851029026200793410340801347, 10.16023593099286492567890694228, 11.64756999248910381551003587622, 12.76116241358441069395324908370, 14.73516815278828022509321103499, 15.62326160682272491601921668065, 16.31559803319823114247811077218
