| L(s) = 1 | + (2 + 2.23i)3-s − 2.23i·5-s + 6·7-s + (−1.00 + 8.94i)9-s + 4.47i·11-s + 16·13-s + (5.00 − 4.47i)15-s − 4.47i·17-s + 2·19-s + (12 + 13.4i)21-s + 13.4i·23-s − 5.00·25-s + (−22.0 + 15.6i)27-s + 31.3i·29-s + 18·31-s + ⋯ |
| L(s) = 1 | + (0.666 + 0.745i)3-s − 0.447i·5-s + 0.857·7-s + (−0.111 + 0.993i)9-s + 0.406i·11-s + 1.23·13-s + (0.333 − 0.298i)15-s − 0.263i·17-s + 0.105·19-s + (0.571 + 0.638i)21-s + 0.583i·23-s − 0.200·25-s + (−0.814 + 0.579i)27-s + 1.07i·29-s + 0.580·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.00551 + 0.766039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00551 + 0.766039i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2 - 2.23i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 - 6T + 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 - 16T + 169T^{2} \) |
| 17 | \( 1 + 4.47iT - 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 - 13.4iT - 529T^{2} \) |
| 29 | \( 1 - 31.3iT - 841T^{2} \) |
| 31 | \( 1 - 18T + 961T^{2} \) |
| 37 | \( 1 + 16T + 1.36e3T^{2} \) |
| 41 | \( 1 + 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16T + 1.84e3T^{2} \) |
| 47 | \( 1 + 49.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.47iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24T + 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74T + 5.32e3T^{2} \) |
| 79 | \( 1 + 138T + 6.24e3T^{2} \) |
| 83 | \( 1 + 93.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 166T + 9.40e3T^{2} \) |
| 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
−11.86920910629561101145235688080, −10.96337369704640198288128117707, −10.07557547052335808002722409524, −8.923208519302524341099040742225, −8.384563143636186599169378747887, −7.25607506890678589335485944452, −5.54549589411894565026202764813, −4.59546880608982404806472075166, −3.45736836724549910421010146035, −1.72511460261424381487328640801,
1.33018138899536871613497848073, 2.81335997239940769447514794145, 4.13586751584988462759068338316, 5.88229745292187659101033738958, 6.80005510706424155515445199127, 8.076471089702827309964779763516, 8.477825762438090264089948587370, 9.798378896442379803191075966203, 11.05596166713077925436426173029, 11.69504423317623663875070676695
