| L(s) = 1 | + (32 + 55.4i)2-s + (−194. + 336. i)3-s + (−2.04e3 + 3.54e3i)4-s + (1.29e4 + 2.24e4i)5-s − 2.48e4·6-s − 2.62e5·8-s + (7.21e5 + 1.24e6i)9-s + (−8.28e5 + 1.43e6i)10-s + (1.12e5 − 1.95e5i)11-s + (−7.96e5 − 1.37e6i)12-s − 2.48e7·13-s − 1.00e7·15-s + (−8.38e6 − 1.45e7i)16-s + (6.10e7 − 1.05e8i)17-s + (−4.61e7 + 7.99e7i)18-s + (−1.07e8 − 1.85e8i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.153 + 0.266i)3-s + (−0.249 + 0.433i)4-s + (0.370 + 0.642i)5-s − 0.217·6-s − 0.353·8-s + (0.452 + 0.783i)9-s + (−0.262 + 0.453i)10-s + (0.0191 − 0.0332i)11-s + (−0.0769 − 0.133i)12-s − 1.42·13-s − 0.228·15-s + (−0.125 − 0.216i)16-s + (0.613 − 1.06i)17-s + (−0.320 + 0.554i)18-s + (−0.522 − 0.905i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.5602407913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5602407913\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + (194. - 336. i)T + (-7.97e5 - 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-1.29e4 - 2.24e4i)T + (-6.10e8 + 1.05e9i)T^{2} \) |
| 11 | \( 1 + (-1.12e5 + 1.95e5i)T + (-1.72e13 - 2.98e13i)T^{2} \) |
| 13 | \( 1 + 2.48e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-6.10e7 + 1.05e8i)T + (-4.95e15 - 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.07e8 + 1.85e8i)T + (-2.10e16 + 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-4.90e8 - 8.50e8i)T + (-2.52e17 + 4.36e17i)T^{2} \) |
| 29 | \( 1 + 2.47e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + (2.44e9 - 4.22e9i)T + (-1.22e19 - 2.11e19i)T^{2} \) |
| 37 | \( 1 + (1.35e10 + 2.33e10i)T + (-1.21e20 + 2.10e20i)T^{2} \) |
| 41 | \( 1 + 3.27e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.78e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-7.58e9 - 1.31e10i)T + (-2.73e21 + 4.72e21i)T^{2} \) |
| 53 | \( 1 + (1.96e10 - 3.40e10i)T + (-1.30e22 - 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-7.22e10 + 1.25e11i)T + (-5.24e22 - 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-7.88e10 - 1.36e11i)T + (-8.09e22 + 1.40e23i)T^{2} \) |
| 67 | \( 1 + (-3.83e11 + 6.64e11i)T + (-2.74e23 - 4.74e23i)T^{2} \) |
| 71 | \( 1 + 1.73e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-7.94e11 + 1.37e12i)T + (-8.35e23 - 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-1.43e11 - 2.47e11i)T + (-2.33e24 + 4.04e24i)T^{2} \) |
| 83 | \( 1 - 9.41e10T + 8.87e24T^{2} \) |
| 89 | \( 1 + (1.97e12 + 3.42e12i)T + (-1.09e25 + 1.90e25i)T^{2} \) |
| 97 | \( 1 - 1.04e13T + 6.73e25T^{2} \) |
| 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
−11.15015420923494615481320946512, −10.10401654222762026038959160931, −9.127008293546849848048510319961, −7.47534678044915116859680647113, −6.99936400215414663944620125785, −5.42968013031748656774390759356, −4.75540569274891704690554258009, −3.20834207200289505279463318006, −2.06275651947543051805763029151, −0.10796149971287072367514104415,
1.12228600566929364228175615608, 2.03966915316363760253719505191, 3.53138980688957487856532561267, 4.70432270817864084885847589643, 5.78017019771074272478903066486, 6.98290469085161978717600605908, 8.457001437239465727703143740801, 9.630306154216794955657084104038, 10.36200237283265824747880227043, 11.83266453574846827016238174357
