| L(s) = 1 | + (−3.23 + 2.35i)2-s + (2.33 − 7.18i)3-s + (4.94 − 15.2i)4-s + (−54.1 + 13.8i)5-s + (9.33 + 28.7i)6-s + 54.5·7-s + (19.7 + 60.8i)8-s + (150. + 109. i)9-s + (142. − 172. i)10-s + (−49.4 + 35.9i)11-s + (−97.7 − 71.0i)12-s + (436. + 316. i)13-s + (−176. + 128. i)14-s + (−26.9 + 421. i)15-s + (−207. − 150. i)16-s + (575. + 1.77e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.149 − 0.460i)3-s + (0.154 − 0.475i)4-s + (−0.968 + 0.247i)5-s + (0.105 + 0.325i)6-s + 0.421·7-s + (0.109 + 0.336i)8-s + (0.619 + 0.449i)9-s + (0.451 − 0.544i)10-s + (−0.123 + 0.0895i)11-s + (−0.195 − 0.142i)12-s + (0.715 + 0.520i)13-s + (−0.240 + 0.175i)14-s + (−0.0308 + 0.483i)15-s + (−0.202 − 0.146i)16-s + (0.482 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.893502 + 0.647846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.893502 + 0.647846i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 5 | \( 1 + (54.1 - 13.8i)T \) |
| good | 3 | \( 1 + (-2.33 + 7.18i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 - 54.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + (49.4 - 35.9i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-436. - 316. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-575. - 1.77e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-590. - 1.81e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.55e3 - 1.85e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-1.05e3 + 3.25e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.64e3 + 5.05e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-8.62e3 - 6.26e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (2.12e3 + 1.54e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.59e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.11e3 + 3.43e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.10e4 - 3.41e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.16e4 + 8.49e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-8.84e3 + 6.42e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-7.82e3 - 2.40e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.68e4 + 5.19e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.97e4 - 1.43e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.02e4 + 3.16e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-3.00e4 - 9.25e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (1.01e5 - 7.38e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-3.88e4 + 1.19e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.95804548034323808736865605069, −13.80630494625335905041937626810, −12.37755121340648176090585995947, −11.18983710503970591132161388909, −9.999445673049966765804977320908, −8.161952800640884192854453617094, −7.71158872225442470599856569773, −6.15914088374765918550768664178, −4.07447304493316266986985198337, −1.54927416741965375993712045350,
0.75300585571866072195584200975, 3.27330047831288987793524281032, 4.73346361233709691608322217834, 7.13575332555102304250655364877, 8.352668166537947494557199884752, 9.473810476858924450112191668302, 10.79978713306574926584110158967, 11.77878881826718126829097824473, 12.88924218322719988069383027226, 14.48387937083838483355493617480
