L(s) = 1 | + 2-s + (1.58 − 2.75i)3-s + 4-s + (−0.5 + 0.866i)5-s + (1.58 − 2.75i)6-s + (1 + 1.73i)7-s + 8-s + (−3.54 − 6.14i)9-s + (−0.5 + 0.866i)10-s + 2.17·11-s + (1.58 − 2.75i)12-s + (−1.45 − 2.52i)13-s + (1 + 1.73i)14-s + (1.58 + 2.75i)15-s + 16-s + (−0.629 − 1.09i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.917 − 1.58i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.648 − 1.12i)6-s + (0.377 + 0.654i)7-s + 0.353·8-s + (−1.18 − 2.04i)9-s + (−0.158 + 0.273i)10-s + 0.656·11-s + (0.458 − 0.794i)12-s + (−0.404 − 0.701i)13-s + (0.267 + 0.462i)14-s + (0.410 + 0.710i)15-s + 0.250·16-s + (−0.152 − 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.24608 - 1.47718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24608 - 1.47718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (-5.31 + 3.84i)T \) |
good | 3 | \( 1 + (-1.58 + 2.75i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 + (1.45 + 2.52i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.629 + 1.09i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 - 4.90i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.17 - 2.03i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.32 - 5.76i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.459 + 0.795i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.71 - 4.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 + (-1.74 + 3.01i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 9.45T + 59T^{2} \) |
| 61 | \( 1 + (-7.27 - 12.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.87 + 10.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.63 - 8.03i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.26 - 12.5i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.35 + 7.54i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.50 - 4.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.758 + 1.31i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−11.49702218258341706314298186701, −10.14059951729944263860372604044, −8.736303937538501626712001618228, −8.141163649351628041708658948932, −7.17922865315738166218133188849, −6.48678669410498083769602676516, −5.43244717468412387581367709296, −3.69320168303290062279327275507, −2.68398514793596120131599428859, −1.62629860491328636587628533642,
2.32863623318681954082416919656, 3.71441962767282644689382266324, 4.38581545992573467743738022597, 4.97456076069443315554960529273, 6.55853580602746068549537594070, 7.83956965042114406127359705871, 8.760842848466674982245954452597, 9.503843427358388061501701149611, 10.49016392840810500215356246349, 11.15467495409071656888059544953