| L(s) = 1 | + (−32 − 55.4i)2-s + (301. − 523. i)3-s + (−2.04e3 + 3.54e3i)4-s + (2.52e3 + 4.36e3i)5-s − 3.86e4·6-s + (3.02e5 − 7.52e4i)7-s + 2.62e5·8-s + (6.14e5 + 1.06e6i)9-s + (1.61e5 − 2.79e5i)10-s + (5.42e5 − 9.38e5i)11-s + (1.23e6 + 2.14e6i)12-s + 4.14e6·13-s + (−1.38e7 − 1.43e7i)14-s + 3.04e6·15-s + (−8.38e6 − 1.45e7i)16-s + (7.24e7 − 1.25e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.239 − 0.414i)3-s + (−0.249 + 0.433i)4-s + (0.0721 + 0.125i)5-s − 0.338·6-s + (0.970 − 0.241i)7-s + 0.353·8-s + (0.385 + 0.667i)9-s + (0.0510 − 0.0884i)10-s + (0.0922 − 0.159i)11-s + (0.119 + 0.207i)12-s + 0.238·13-s + (−0.491 − 0.508i)14-s + 0.0690·15-s + (−0.125 − 0.216i)16-s + (0.728 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.43033 - 1.19258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43033 - 1.19258i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (32 + 55.4i)T \) |
| 7 | \( 1 + (-3.02e5 + 7.52e4i)T \) |
| good | 3 | \( 1 + (-301. + 523. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-2.52e3 - 4.36e3i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-5.42e5 + 9.38e5i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 - 4.14e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-7.24e7 + 1.25e8i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.39e8 + 2.41e8i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (4.18e7 + 7.25e7i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 - 6.32e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + (1.65e9 - 2.87e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (-8.76e9 - 1.51e10i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 - 1.15e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.19e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.47e10 + 4.28e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-1.09e11 + 1.89e11i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (6.67e10 - 1.15e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-6.84e10 - 1.18e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (4.61e11 - 7.99e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + 9.00e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + (9.23e11 - 1.59e12i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (1.46e12 + 2.53e12i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 - 4.12e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + (2.20e12 + 3.81e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 - 3.35e12T + 6.73e25T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.28231351849401552132664839883, −14.36003440123387689860160047222, −13.23336999376398239431629945865, −11.63742688887621372956834147745, −10.40760862771394139425117366906, −8.596820644764143442176110871480, −7.27292255451508168381304373020, −4.70834082828787489641643198600, −2.51517502295507405607491982020, −0.994437450618950455823993121946,
1.41975547149108629559314415842, 4.10061134584600035771671004563, 5.89733127094827693787108809787, 7.84172229070845956759314329917, 9.146455061108174495024850644256, 10.62535120057788951332036743896, 12.50745981187785423019486515767, 14.47157762067657935910685117069, 15.16588297110151605472091204005, 16.67166743141026916463618247198
