| L(s) = 1 | + 28.7·5-s − 80.4·7-s − 40.7·11-s + 36.7i·13-s − 108. i·17-s + 264. i·19-s − 406. i·23-s + 203.·25-s + 805.·29-s − 1.43e3·31-s − 2.31e3·35-s + 910. i·37-s − 2.44e3i·41-s + 3.54e3i·43-s − 1.74e3i·47-s + ⋯ |
| L(s) = 1 | + 1.15·5-s − 1.64·7-s − 0.336·11-s + 0.217i·13-s − 0.374i·17-s + 0.732i·19-s − 0.768i·23-s + 0.326·25-s + 0.957·29-s − 1.49·31-s − 1.89·35-s + 0.665i·37-s − 1.45i·41-s + 1.91i·43-s − 0.787i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.818544733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818544733\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 28.7T + 625T^{2} \) |
| 7 | \( 1 + 80.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 40.7T + 1.46e4T^{2} \) |
| 13 | \( 1 - 36.7iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 108. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 264. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 406. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 805.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.43e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 910. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.44e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.54e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.74e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.36e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 4.58e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.30e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.69e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.38e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.79e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.80e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.47e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 2.39e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.35e4T + 8.85e7T^{2} \) |
| 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
−9.327411393200032453681602626423, −8.662729802023582857088306453943, −7.39564551964102354110503917100, −6.53245177164372298516208914314, −6.00194449431834282168479401391, −5.14190658210233589304892404797, −3.82864476403994310036750207899, −2.87689053943552764360817440574, −1.99356947278581190727422256647, −0.56563467650881831503119390037,
0.63839220715025276201693335932, 2.05189768488990202815206905038, 2.93970121840643897898121239983, 3.86068608399400530417861191597, 5.28351881217471272951601193539, 5.90506399601713715689612166221, 6.66380002384944422786683157252, 7.42097465272404389051995409025, 8.743541043252175219366420076833, 9.353123723008106223895121754463
