| L(s) = 1 | + (−0.490 + 0.343i)2-s + (−0.561 + 1.54i)4-s + (1.25 − 1.85i)5-s + (−0.0636 + 0.136i)7-s + (−0.563 − 2.10i)8-s + (0.0215 + 1.33i)10-s + (3.54 − 4.23i)11-s + (1.08 − 1.54i)13-s + (−0.0156 − 0.0887i)14-s + (−1.51 − 1.27i)16-s + (−1.18 + 4.40i)17-s + (6.55 + 3.78i)19-s + (2.15 + 2.97i)20-s + (−0.287 + 3.29i)22-s + (0.883 − 0.412i)23-s + ⋯ |
| L(s) = 1 | + (−0.346 + 0.242i)2-s + (−0.280 + 0.771i)4-s + (0.560 − 0.828i)5-s + (−0.0240 + 0.0515i)7-s + (−0.199 − 0.744i)8-s + (0.00682 + 0.422i)10-s + (1.07 − 1.27i)11-s + (0.300 − 0.429i)13-s + (−0.00418 − 0.0237i)14-s + (−0.379 − 0.318i)16-s + (−0.286 + 1.06i)17-s + (1.50 + 0.867i)19-s + (0.481 + 0.664i)20-s + (−0.0613 + 0.701i)22-s + (0.184 − 0.0859i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23765 + 0.00882271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23765 + 0.00882271i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.25 + 1.85i)T \) |
| good | 2 | \( 1 + (0.490 - 0.343i)T + (0.684 - 1.87i)T^{2} \) |
| 7 | \( 1 + (0.0636 - 0.136i)T + (-4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (-3.54 + 4.23i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.08 + 1.54i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.18 - 4.40i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.55 - 3.78i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.883 + 0.412i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.486 + 2.75i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.70 - 0.622i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.71 - 1.53i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.682 + 0.120i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.151 - 1.73i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (10.0 + 4.70i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.204 - 0.204i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.796 - 0.668i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-9.12 + 3.32i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (8.96 + 6.27i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-2.53 + 1.46i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (9.95 - 2.66i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (11.9 + 2.10i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (4.10 + 5.85i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (6.74 - 11.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.56 + 0.311i)T + (95.5 - 16.8i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42518956146172639255955110145, −10.06559703637475310336063588997, −9.242402017576539197567051242555, −8.498211670489575281895406316247, −7.893473389019069779260980298312, −6.44479441984948839587579110212, −5.66822228788414907000826409535, −4.22493475591963686822702465439, −3.23212192833490314343684589742, −1.14124257505695039024895091163,
1.44122318849848736086234146679, 2.75539665115150677603817812843, 4.43545571530584795390905808325, 5.48870444733756214820374838977, 6.67643992240054295827466821693, 7.26616570859610662831841963199, 8.970665713636828031630126704415, 9.573865705024676735664916932664, 10.09340827070687355184646572957, 11.33229497018999057678776208489
