| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−1.14 + 1.29i)3-s + (0.913 + 0.406i)4-s + (−1.90 − 1.17i)5-s + (1.39 − 1.02i)6-s + (−2.23 + 3.87i)7-s + (−0.809 − 0.587i)8-s + (−0.357 − 2.97i)9-s + (1.61 + 1.54i)10-s + (4.32 + 0.920i)11-s + (−1.57 + 0.716i)12-s + (−1.76 + 0.376i)13-s + (2.99 − 3.32i)14-s + (3.71 − 1.11i)15-s + (0.669 + 0.743i)16-s + (−4.08 − 2.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.663 + 0.748i)3-s + (0.456 + 0.203i)4-s + (−0.850 − 0.526i)5-s + (0.568 − 0.419i)6-s + (−0.846 + 1.46i)7-s + (−0.286 − 0.207i)8-s + (−0.119 − 0.992i)9-s + (0.510 + 0.488i)10-s + (1.30 + 0.277i)11-s + (−0.455 + 0.206i)12-s + (−0.490 + 0.104i)13-s + (0.801 − 0.889i)14-s + (0.957 − 0.286i)15-s + (0.167 + 0.185i)16-s + (−0.990 − 0.719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0532 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0532 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.174738 - 0.184298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174738 - 0.184298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (1.14 - 1.29i)T \) |
| 5 | \( 1 + (1.90 + 1.17i)T \) |
| good | 7 | \( 1 + (2.23 - 3.87i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.32 - 0.920i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.376i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (4.08 + 2.96i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.86 + 2.80i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-5.23 + 5.81i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.130 + 1.23i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 2.94i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.22 + 9.92i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (3.40 - 0.724i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.851 - 1.47i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.09 + 10.3i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (1.31 - 0.953i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (10.5 - 2.24i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.24 - 0.689i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.106 - 1.01i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-4.74 + 3.44i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.31 - 7.11i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.425 - 4.05i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (6.72 - 2.99i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (3.47 + 10.6i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.402 - 3.82i)T + (-94.8 + 20.1i)T^{2} \) |
| 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
−11.02651317845317895242535067491, −9.712013358993494979188274689358, −8.988159639886065538184763996936, −8.760011252070858679536266862465, −6.97971249281370994580178925952, −6.32773771664753830969715899919, −5.00310045226809939758477991014, −4.01744110052420198621971825906, −2.59732961181146599294902657724, −0.23455720974965389481511519505,
1.29517979642754311621225300946, 3.31135368108027642333285599703, 4.45492372658227129221021463555, 6.35662290413366771493744269639, 6.71709700403100126738676905044, 7.46987015888330866094518851372, 8.380757462687398769620464422165, 9.640840894699498385744550929112, 10.67699700840960227959896678414, 11.08257160875736966926959016530
