| L(s) = 1 | + (−0.463 − 0.803i)2-s + (−3.61 − 6.26i)3-s + (63.5 − 110. i)4-s − 525.·5-s + (−3.35 + 5.80i)6-s + (171.5 − 297. i)7-s − 236.·8-s + (1.06e3 − 1.84e3i)9-s + (243. + 422. i)10-s + (−3.05e3 − 5.29e3i)11-s − 919.·12-s + (−35.4 + 7.92e3i)13-s − 318.·14-s + (1.89e3 + 3.29e3i)15-s + (−8.02e3 − 1.39e4i)16-s + (6.12e3 − 1.06e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0409 − 0.0709i)2-s + (−0.0772 − 0.133i)3-s + (0.496 − 0.860i)4-s − 1.88·5-s + (−0.00633 + 0.0109i)6-s + (0.188 − 0.327i)7-s − 0.163·8-s + (0.488 − 0.845i)9-s + (0.0770 + 0.133i)10-s + (−0.692 − 1.19i)11-s − 0.153·12-s + (−0.00447 + 0.999i)13-s − 0.0309·14-s + (0.145 + 0.251i)15-s + (−0.489 − 0.848i)16-s + (0.302 − 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0320580 + 0.0418627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0320580 + 0.0418627i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (35.4 - 7.92e3i)T \) |
| good | 2 | \( 1 + (0.463 + 0.803i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (3.61 + 6.26i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 525.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (3.05e3 + 5.29e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-6.12e3 + 1.06e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.03e4 - 3.52e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.79e3 - 1.17e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.67e4 + 4.64e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 8.46e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.38e5 - 2.40e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-3.92e5 - 6.79e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (2.27e5 - 3.93e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 5.67e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.21e5 + 7.30e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.25e5 + 3.89e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.66e6 - 2.87e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-8.12e5 + 1.40e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 1.27e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.85e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.69e6 + 8.13e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.26e5 + 2.18e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−11.50758903272355440139546606047, −11.26588810283945947950077608906, −9.860330860704211146703614327465, −8.355840926651988097802533128144, −7.33367330042238602559127277965, −6.21946147206684854047435757252, −4.51028518445846194986289086359, −3.29824493155620309499116732935, −1.14170714946665739284371860559, −0.01933839178053537647607395299,
2.43556741102744330153125059072, 3.84468908469672309130939857147, 4.91658985160902813538575129166, 7.18567081634848774682553417479, 7.69051277338373084932455005699, 8.586211116532947375692368584486, 10.58856261061074831840929869056, 11.27214440563847299569497549738, 12.56938988141012457551115857179, 12.77338690510433503048423590471
