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
−11.15467495409071656888059544953, −10.49016392840810500215356246349, −9.503843427358388061501701149611, −8.760842848466674982245954452597, −7.83956965042114406127359705871, −6.55853580602746068549537594070, −4.97456076069443315554960529273, −4.38581545992573467743738022597, −3.71441962767282644689382266324, −2.32863623318681954082416919656,
1.62629860491328636587628533642, 2.68398514793596120131599428859, 3.69320168303290062279327275507, 5.43244717468412387581367709296, 6.48678669410498083769602676516, 7.17922865315738166218133188849, 8.141163649351628041708658948932, 8.736303937538501626712001618228, 10.14059951729944263860372604044, 11.49702218258341706314298186701