| L(s) = 1 | + (−0.607 − 0.791i)2-s + (0.259 − 0.970i)4-s + (3.76 + 0.246i)5-s + (−0.260 − 3.97i)7-s + (−2.76 + 1.14i)8-s + (−2.08 − 3.12i)10-s + (−2.97 − 2.60i)11-s + (0.149 + 0.0401i)13-s + (−2.99 + 2.62i)14-s + (0.850 + 0.490i)16-s + (1.50 + 3.83i)17-s + (3.79 + 1.57i)19-s + (1.21 − 3.58i)20-s + (−0.257 + 3.93i)22-s + (−0.314 − 0.926i)23-s + ⋯ |
| L(s) = 1 | + (−0.429 − 0.559i)2-s + (0.129 − 0.485i)4-s + (1.68 + 0.110i)5-s + (−0.0985 − 1.50i)7-s + (−0.979 + 0.405i)8-s + (−0.660 − 0.988i)10-s + (−0.896 − 0.785i)11-s + (0.0415 + 0.0111i)13-s + (−0.799 + 0.701i)14-s + (0.212 + 0.122i)16-s + (0.364 + 0.931i)17-s + (0.870 + 0.360i)19-s + (0.272 − 0.801i)20-s + (−0.0549 + 0.838i)22-s + (−0.0656 − 0.193i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.745496 - 1.16090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.745496 - 1.16090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + (-1.50 - 3.83i)T \) |
| good | 2 | \( 1 + (0.607 + 0.791i)T + (-0.517 + 1.93i)T^{2} \) |
| 5 | \( 1 + (-3.76 - 0.246i)T + (4.95 + 0.652i)T^{2} \) |
| 7 | \( 1 + (0.260 + 3.97i)T + (-6.94 + 0.913i)T^{2} \) |
| 11 | \( 1 + (2.97 + 2.60i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.149 - 0.0401i)T + (11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-3.79 - 1.57i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.314 + 0.926i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (4.46 - 2.20i)T + (17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (3.23 + 3.69i)T + (-4.04 + 30.7i)T^{2} \) |
| 37 | \( 1 + (0.326 + 1.64i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-2.15 + 4.36i)T + (-24.9 - 32.5i)T^{2} \) |
| 43 | \( 1 + (0.517 - 3.93i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (-2.97 + 0.797i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.152 + 0.367i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.37 + 1.05i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-7.89 + 0.517i)T + (60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (-9.67 + 5.58i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.55 + 0.707i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-12.4 - 8.29i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (7.85 - 8.95i)T + (-10.3 - 78.3i)T^{2} \) |
| 83 | \( 1 + (-0.349 + 0.268i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (3.79 - 3.79i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.20 + 6.50i)T + (-59.0 + 76.9i)T^{2} \) |
| 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
−10.70843788700960478799612303081, −9.948882330318358828680172495136, −9.466196348311752588541843030491, −8.207243647128174680828899068048, −6.98251111561285182083072264579, −5.91563852088337745484381772877, −5.35609992179906761386582783428, −3.54414323045709348641906192131, −2.19651525132149663682228145153, −1.01475593330780914343573469588,
2.15929869046731586112724158916, 2.94398241934589535532623886132, 5.20142780575706442716008058480, 5.66559255700379353833238016843, 6.72965941936423995272367881376, 7.68953694596197624720812454653, 8.831093096101429693999334665755, 9.422558490706491975978440576319, 9.981780859939750069060418313106, 11.42833057903125142896568214022
