| L(s) = 1 | + (−19.7 + 60.8i)2-s + (−351. + 255. i)3-s + (−3.31e3 − 2.40e3i)4-s + (−3.08e3 − 9.50e3i)5-s + (−8.59e3 − 2.64e4i)6-s + (3.27e5 + 2.37e5i)7-s + (2.12e5 − 1.54e5i)8-s + (−4.34e5 + 1.33e6i)9-s + 6.39e5·10-s + (−5.87e6 + 1.66e5i)11-s + 1.78e6·12-s + (−1.05e6 + 3.24e6i)13-s + (−2.09e7 + 1.52e7i)14-s + (3.51e6 + 2.55e6i)15-s + (5.18e6 + 1.59e7i)16-s + (−5.11e6 − 1.57e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.278 + 0.202i)3-s + (−0.404 − 0.293i)4-s + (−0.0883 − 0.272i)5-s + (−0.0752 − 0.231i)6-s + (1.05 + 0.764i)7-s + (0.286 − 0.207i)8-s + (−0.272 + 0.838i)9-s + 0.202·10-s + (−0.999 + 0.0283i)11-s + 0.172·12-s + (−0.0605 + 0.186i)13-s + (−0.743 + 0.540i)14-s + (0.0796 + 0.0578i)15-s + (0.0772 + 0.237i)16-s + (−0.0514 − 0.158i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.1657656542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1657656542\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.7 - 60.8i)T \) |
| 11 | \( 1 + (5.87e6 - 1.66e5i)T \) |
| good | 3 | \( 1 + (351. - 255. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (3.08e3 + 9.50e3i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-3.27e5 - 2.37e5i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (1.05e6 - 3.24e6i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (5.11e6 + 1.57e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (3.12e7 - 2.27e7i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 + 2.88e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (3.78e9 + 2.74e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-1.04e9 + 3.22e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (2.33e10 + 1.69e10i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (4.51e10 - 3.27e10i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 + 3.66e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-2.33e10 + 1.69e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (6.32e9 - 1.94e10i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (6.06e10 + 4.40e10i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-1.90e11 - 5.87e11i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 + 8.64e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.44e10 + 4.43e10i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (4.28e11 + 3.11e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (1.67e11 - 5.13e11i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (1.64e12 + 5.07e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 - 4.25e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.06e12 - 3.27e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−15.80449044181134738071578303256, −14.72275352280331492915126169806, −13.37987734891141943683020308454, −11.72459538333486009215773587377, −10.41567688828252696931097173415, −8.668492993990208695245354928262, −7.70435859690344315391740400434, −5.65118685206428402623739941171, −4.73455727381190541426589095417, −2.08951349198595371749879251094,
0.06247311654181676126679468971, 1.52805847518911972948262133652, 3.38597124149789011906745731280, 5.10891068124002802755474146165, 7.17153079372138911653326976413, 8.567308644080056560022993202979, 10.36066451116017767837300625721, 11.25862203133019698650893638467, 12.53610006972028405182046181060, 13.91168528154903596754109798788
