| L(s) = 1 | + (0.200 − 1.90i)2-s + (−1.50 + 1.41i)3-s + (−1.64 − 0.350i)4-s + (−3.54 + 1.76i)5-s + (2.39 + 3.15i)6-s + (−1.71 + 3.27i)7-s + (0.187 − 0.577i)8-s + (0.0794 − 1.26i)9-s + (2.64 + 7.12i)10-s + (0.717 − 3.05i)11-s + (2.97 − 1.80i)12-s + (−2.96 − 0.248i)13-s + (5.90 + 3.92i)14-s + (2.85 − 7.66i)15-s + (−4.14 − 1.84i)16-s + (−0.0397 − 0.00166i)17-s + ⋯ |
| L(s) = 1 | + (0.141 − 1.34i)2-s + (−0.869 + 0.816i)3-s + (−0.823 − 0.175i)4-s + (−1.58 + 0.787i)5-s + (0.978 + 1.28i)6-s + (−0.646 + 1.23i)7-s + (0.0663 − 0.204i)8-s + (0.0264 − 0.421i)9-s + (0.837 + 2.25i)10-s + (0.216 − 0.922i)11-s + (0.858 − 0.519i)12-s + (−0.822 − 0.0690i)13-s + (1.57 + 1.04i)14-s + (0.735 − 1.97i)15-s + (−1.03 − 0.460i)16-s + (−0.00963 − 0.000403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00709 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00709 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.243061 + 0.244790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243061 + 0.244790i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-5.82 + 10.8i)T \) |
| good | 2 | \( 1 + (-0.200 + 1.90i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (1.50 - 1.41i)T + (0.188 - 2.99i)T^{2} \) |
| 5 | \( 1 + (3.54 - 1.76i)T + (3.02 - 3.98i)T^{2} \) |
| 7 | \( 1 + (1.71 - 3.27i)T + (-3.99 - 5.74i)T^{2} \) |
| 11 | \( 1 + (-0.717 + 3.05i)T + (-9.85 - 4.89i)T^{2} \) |
| 13 | \( 1 + (2.96 + 0.248i)T + (12.8 + 2.16i)T^{2} \) |
| 17 | \( 1 + (0.0397 + 0.00166i)T + (16.9 + 1.42i)T^{2} \) |
| 19 | \( 1 + (-2.13 - 6.58i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.529 + 0.587i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 7.37i)T + (-26.9 + 10.6i)T^{2} \) |
| 31 | \( 1 + (-0.618 - 0.490i)T + (7.07 + 30.1i)T^{2} \) |
| 37 | \( 1 + (4.57 - 6.57i)T + (-12.8 - 34.6i)T^{2} \) |
| 41 | \( 1 + (3.38 - 0.427i)T + (39.7 - 10.1i)T^{2} \) |
| 43 | \( 1 + (0.00513 + 0.00981i)T + (-24.5 + 35.3i)T^{2} \) |
| 47 | \( 1 + (-1.58 + 3.75i)T + (-32.8 - 33.5i)T^{2} \) |
| 53 | \( 1 + (8.31 - 4.57i)T + (28.3 - 44.7i)T^{2} \) |
| 59 | \( 1 + (-9.85 + 7.16i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.35 - 1.01i)T + (50.8 - 33.7i)T^{2} \) |
| 67 | \( 1 + (0.00118 + 0.0188i)T + (-66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-12.9 + 0.544i)T + (70.7 - 5.94i)T^{2} \) |
| 73 | \( 1 + (-7.81 - 12.3i)T + (-31.0 + 66.0i)T^{2} \) |
| 79 | \( 1 + (-0.356 - 0.141i)T + (57.5 + 54.0i)T^{2} \) |
| 83 | \( 1 + (5.75 + 1.47i)T + (72.7 + 39.9i)T^{2} \) |
| 89 | \( 1 + (3.78 - 3.86i)T + (-1.86 - 88.9i)T^{2} \) |
| 97 | \( 1 + (2.26 - 1.50i)T + (37.5 - 89.4i)T^{2} \) |
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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
−12.45488109696755071379655056795, −12.02233337102808238745385389490, −11.33757454131717841246107921714, −10.58158179117576576820483511722, −9.717326718098120428929438138918, −8.288704348003053071245002688966, −6.70317689956140488768737706932, −5.21933104101225062213222549001, −3.78951859822933072284740282698, −2.98294852720673203552784650281,
0.35003714653336575331877285890, 4.16643757383522322604243758571, 5.10014441312678951950616652934, 6.76888061352508818563835092316, 7.18090846840208842974579944126, 7.897341141430482784984352837842, 9.379361825200869927310859788394, 11.09183847906643899301458953038, 11.95268218021193946972960003239, 12.76747622662682362796110771807
