| L(s) = 1 | + (−0.212 + 0.977i)2-s + (−0.800 + 0.599i)3-s + (−0.909 − 0.415i)4-s + (−2.06 + 0.859i)5-s + (−0.415 − 0.909i)6-s + (−0.0521 + 0.729i)7-s + (0.599 − 0.800i)8-s + (0.281 − 0.959i)9-s + (−0.400 − 2.19i)10-s + (−2.88 + 4.48i)11-s + (0.977 − 0.212i)12-s + (−1.14 + 0.0820i)13-s + (−0.701 − 0.206i)14-s + (1.13 − 1.92i)15-s + (0.654 + 0.755i)16-s + (1.44 − 3.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.690i)2-s + (−0.462 + 0.345i)3-s + (−0.454 − 0.207i)4-s + (−0.923 + 0.384i)5-s + (−0.169 − 0.371i)6-s + (−0.0197 + 0.275i)7-s + (0.211 − 0.283i)8-s + (0.0939 − 0.319i)9-s + (−0.126 − 0.695i)10-s + (−0.869 + 1.35i)11-s + (0.282 − 0.0613i)12-s + (−0.318 + 0.0227i)13-s + (−0.187 − 0.0550i)14-s + (0.293 − 0.497i)15-s + (0.163 + 0.188i)16-s + (0.349 − 0.938i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.123317 - 0.0886632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.123317 - 0.0886632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.212 - 0.977i)T \) |
| 3 | \( 1 + (0.800 - 0.599i)T \) |
| 5 | \( 1 + (2.06 - 0.859i)T \) |
| 23 | \( 1 + (4.73 + 0.732i)T \) |
| good | 7 | \( 1 + (0.0521 - 0.729i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (2.88 - 4.48i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.14 - 0.0820i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 3.86i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 3.04i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.942 + 0.430i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.548 + 3.81i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (4.89 + 8.97i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (2.29 - 0.672i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.27 - 1.69i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-5.98 + 5.98i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.56 + 0.541i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-3.23 - 2.80i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-6.71 + 0.965i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (9.98 + 2.17i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (1.93 - 1.24i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (8.25 - 3.07i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-0.664 + 0.767i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (13.4 - 7.33i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (1.07 - 7.48i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (9.70 + 5.29i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.16969805019717339744892962706, −9.529184500508344033984490510731, −8.437418937096154747584557979697, −7.34182030101392165474221817099, −7.14118785898650562400158745676, −5.73480745597571370993332297284, −4.87853413248281443134977166100, −4.05638461783346169199657104508, −2.56213278368686520051864000413, −0.096377925925389033385909279895,
1.31530618979908964802586591101, 3.05897710923826270096994938895, 3.97650859670357219733338124807, 5.14272987927705225020236746516, 6.03912755638551713051885725504, 7.41977104468156764173415202885, 8.137160558500403795237333640317, 8.723650659542710790184893015327, 10.17305391143129026772330181064, 10.63015148958994699578591788815
