| L(s) = 1 | + (−0.841 − 0.540i)2-s + (2.59 + 0.760i)3-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−1.76 − 2.04i)6-s + (−1.23 − 0.791i)7-s + (0.142 − 0.989i)8-s + (3.61 + 2.32i)9-s + (0.959 − 0.281i)10-s + (−3.91 + 4.52i)11-s + (0.384 + 2.67i)12-s + (0.472 + 3.28i)13-s + (0.608 + 1.33i)14-s + (−2.27 + 1.46i)15-s + (−0.654 + 0.755i)16-s + (−2.52 + 5.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (1.49 + 0.439i)3-s + (0.207 + 0.454i)4-s + (−0.292 + 0.337i)5-s + (−0.722 − 0.833i)6-s + (−0.465 − 0.299i)7-s + (0.0503 − 0.349i)8-s + (1.20 + 0.773i)9-s + (0.303 − 0.0890i)10-s + (−1.18 + 1.36i)11-s + (0.110 + 0.771i)12-s + (0.131 + 0.912i)13-s + (0.162 + 0.355i)14-s + (−0.586 + 0.376i)15-s + (−0.163 + 0.188i)16-s + (−0.613 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20893 + 0.861689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20893 + 0.861689i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (8.18 + 0.113i)T \) |
| good | 3 | \( 1 + (-2.59 - 0.760i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (1.23 + 0.791i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (3.91 - 4.52i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.472 - 3.28i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.52 - 5.53i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.59 + 1.67i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (-6.07 - 1.78i)T + (19.3 + 12.4i)T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 + (-0.0668 + 0.465i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + 2.95T + 37T^{2} \) |
| 41 | \( 1 + (1.79 - 3.93i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.34 + 7.33i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.27 - 1.25i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (0.742 + 1.62i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.96 + 13.6i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-0.694 - 0.800i)T + (-8.68 + 60.3i)T^{2} \) |
| 71 | \( 1 + (-1.72 - 3.78i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (3.58 + 4.13i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.20 - 15.3i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (8.54 - 9.85i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.70 + 1.67i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
| 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
−10.31491142561283175399021938705, −9.842125256046848111399846800543, −8.972541015944278417770256242679, −8.289216073960301960625926133489, −7.39012771702504740896102146926, −6.75129276791437730092173425219, −4.77293955782638463964630184928, −3.83259973178299853739982985890, −2.87944740975178932085882455848, −1.96891853607959657917942229525,
0.805907122367553431816014396746, 2.76421493050592142301220655460, 3.10011676391107641123014003416, 4.93365098616891305628495975324, 6.00868596935498098985998952447, 7.27780142871366810097417155221, 7.81899718281084248772736874557, 8.759345468018567630701285596345, 8.929047988032651665850874562922, 10.10891630880817403869702195893
