L(s) = 1 | + 5.05i·2-s + (10.0 + 25.0i)3-s + 38.4·4-s + 24.8i·5-s + (−126. + 50.7i)6-s − 129.·7-s + 518. i·8-s + (−527. + 503. i)9-s − 125.·10-s − 230. i·11-s + (385. + 962. i)12-s + 690.·13-s − 655. i·14-s + (−622. + 249. i)15-s − 161.·16-s − 4.80e3i·17-s + ⋯ |
L(s) = 1 | + 0.632i·2-s + (0.371 + 0.928i)3-s + 0.600·4-s + 0.198i·5-s + (−0.586 + 0.235i)6-s − 0.377·7-s + 1.01i·8-s + (−0.723 + 0.690i)9-s − 0.125·10-s − 0.173i·11-s + (0.223 + 0.557i)12-s + 0.314·13-s − 0.238i·14-s + (−0.184 + 0.0738i)15-s − 0.0394·16-s − 0.978i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.04769 + 1.54815i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04769 + 1.54815i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-10.0 - 25.0i)T \) |
| 7 | \( 1 + 129.T \) |
good | 2 | \( 1 - 5.05iT - 64T^{2} \) |
| 5 | \( 1 - 24.8iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 230. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 690.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.80e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.11e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + 8.67e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.42e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 7.15e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.95e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.02e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.08e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 2.03e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 3.51e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.24e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.54e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.23e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 6.22e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 9.99e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.75e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.75e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 6.17e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.14e6T + 8.32e11T^{2} \) |
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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
−16.60165136167440597959739982826, −16.04838504036724622163425771682, −14.91386724504604122274896884340, −13.83601156947243866516557204319, −11.67378652421238297433237551784, −10.38658062324198360837357821661, −8.815181649672686515552723819096, −7.15049047303824441802876202310, −5.35118093214802912043057852106, −3.02622517914671722806383721871,
1.33912215542198745202664543877, 3.15508755600966401002801477697, 6.29145370741519922490429987360, 7.74646694162602800941433637505, 9.591387863896811935883244607436, 11.32559373920816084758486735187, 12.44161819732458404333768873851, 13.45495433633204693359441904609, 15.08343999216347933981497610598, 16.47497571437292266457385953962