| L(s) = 1 | − 4i·2-s + 9i·3-s − 16·4-s + (−49.5 + 25.8i)5-s + 36·6-s − 43.6i·7-s + 64i·8-s − 81·9-s + (103. + 198. i)10-s + 344.·11-s − 144i·12-s − 114. i·13-s − 174.·14-s + (−232. − 446. i)15-s + 256·16-s + 628. i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.886 + 0.462i)5-s + 0.408·6-s − 0.336i·7-s + 0.353i·8-s − 0.333·9-s + (0.326 + 0.626i)10-s + 0.859·11-s − 0.288i·12-s − 0.188i·13-s − 0.238·14-s + (−0.266 − 0.511i)15-s + 0.250·16-s + 0.527i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.266829166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.266829166\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 - 9iT \) |
| 5 | \( 1 + (49.5 - 25.8i)T \) |
| 19 | \( 1 + 361T \) |
| good | 7 | \( 1 + 43.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 344.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 114. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 628. iT - 1.41e6T^{2} \) |
| 23 | \( 1 + 2.21e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.02e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 470.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.17e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.33e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.44e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.35e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.02e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.57e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.96e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e5iT - 8.58e9T^{2} \) |
| 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
−10.43115408066354546408999110642, −9.172553774585705956253527105605, −8.505368615316558625382412623628, −7.43329950613749578636988578853, −6.42850810724601608940782682781, −5.10916570326920827082112766976, −3.94504606998284555787350119944, −3.61590862943364049204400063256, −2.23527937808253120342933238149, −0.71402959242958380543219577238,
0.43661870213616888106353014128, 1.64927416390761477302275029176, 3.33215816920475728127201009799, 4.33988189727595625168742118196, 5.39847272355031620263931992103, 6.36261977333267039929697627784, 7.35406393752989869214471739316, 7.87066599061491726280633215252, 9.014157232933447963052425890518, 9.346470889996816967714627861897
