L(s) = 1 | + (1.11e4 + 1.11e4i)2-s + (1.76e6 − 1.76e6i)3-s + 1.80e8i·4-s + (−1.23e8 + 1.21e9i)5-s + 3.92e10·6-s + (1.33e10 + 1.33e10i)7-s + (−1.25e12 + 1.25e12i)8-s − 3.67e12i·9-s + (−1.48e13 + 1.21e13i)10-s + 9.56e12·11-s + (3.17e14 + 3.17e14i)12-s + (4.32e13 − 4.32e13i)13-s + 2.97e14i·14-s + (1.92e15 + 2.35e15i)15-s − 1.58e16·16-s + (7.63e15 + 7.63e15i)17-s + ⋯ |
L(s) = 1 | + (1.35 + 1.35i)2-s + (1.10 − 1.10i)3-s + 2.68i·4-s + (−0.100 + 0.994i)5-s + 3.00·6-s + (0.138 + 0.138i)7-s + (−2.28 + 2.28i)8-s − 1.44i·9-s + (−1.48 + 1.21i)10-s + 0.276·11-s + (2.96 + 2.96i)12-s + (0.142 − 0.142i)13-s + 0.374i·14-s + (0.988 + 1.21i)15-s − 3.52·16-s + (0.770 + 0.770i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{27}{2})\) |
\(\approx\) |
\(2.90044 + 4.63562i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.90044 + 4.63562i\) |
\(L(14)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.23e8 - 1.21e9i)T \) |
good | 2 | \( 1 + (-1.11e4 - 1.11e4i)T + 6.71e7iT^{2} \) |
| 3 | \( 1 + (-1.76e6 + 1.76e6i)T - 2.54e12iT^{2} \) |
| 7 | \( 1 + (-1.33e10 - 1.33e10i)T + 9.38e21iT^{2} \) |
| 11 | \( 1 - 9.56e12T + 1.19e27T^{2} \) |
| 13 | \( 1 + (-4.32e13 + 4.32e13i)T - 9.17e28iT^{2} \) |
| 17 | \( 1 + (-7.63e15 - 7.63e15i)T + 9.81e31iT^{2} \) |
| 19 | \( 1 + 3.69e15iT - 1.76e33T^{2} \) |
| 23 | \( 1 + (-6.37e16 + 6.37e16i)T - 2.54e35iT^{2} \) |
| 29 | \( 1 + 1.61e19iT - 1.05e38T^{2} \) |
| 31 | \( 1 - 2.82e19T + 5.96e38T^{2} \) |
| 37 | \( 1 + (1.75e20 + 1.75e20i)T + 5.93e40iT^{2} \) |
| 41 | \( 1 + 5.21e20T + 8.55e41T^{2} \) |
| 43 | \( 1 + (-1.20e21 + 1.20e21i)T - 2.95e42iT^{2} \) |
| 47 | \( 1 + (-5.45e21 - 5.45e21i)T + 2.98e43iT^{2} \) |
| 53 | \( 1 + (8.65e20 - 8.65e20i)T - 6.77e44iT^{2} \) |
| 59 | \( 1 + 2.83e22iT - 1.10e46T^{2} \) |
| 61 | \( 1 - 2.33e23T + 2.62e46T^{2} \) |
| 67 | \( 1 + (7.52e23 + 7.52e23i)T + 3.00e47iT^{2} \) |
| 71 | \( 1 + 7.35e23T + 1.35e48T^{2} \) |
| 73 | \( 1 + (7.67e23 - 7.67e23i)T - 2.79e48iT^{2} \) |
| 79 | \( 1 - 5.55e24iT - 2.17e49T^{2} \) |
| 83 | \( 1 + (-3.80e24 + 3.80e24i)T - 7.87e49iT^{2} \) |
| 89 | \( 1 - 5.72e24iT - 4.83e50T^{2} \) |
| 97 | \( 1 + (4.50e25 + 4.50e25i)T + 4.52e51iT^{2} \) |
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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
−17.48872024088933363568776757717, −15.36739489161838156562940323158, −14.40237566711127522439484941793, −13.54127458911636465497284888950, −12.15085352773890772559399255867, −8.276217097464873011197608379052, −7.28132490109756215040783689791, −6.11577838833018512543503015093, −3.72191004958016663294634157281, −2.49792082951214846318648494422,
1.26389029503922792865635226052, 2.95925595035771079118298388244, 4.11362464869111441844919276000, 5.13914544886561342093665266136, 9.023374047447705867764113411889, 10.22702515889826719124609259489, 11.96867954763487714586138002417, 13.54287034164059881731010998939, 14.58353317532009389065586805139, 15.94409864470385458970195104468