L(s) = 1 | + 362. i·2-s + (−1.90e4 + 5.06e3i)3-s − 1.31e5·4-s + 2.98e6i·5-s + (−1.83e6 − 6.88e6i)6-s − 4.92e7·7-s − 4.74e7i·8-s + (3.36e8 − 1.92e8i)9-s − 1.08e9·10-s − 5.85e8i·11-s + (2.49e9 − 6.64e8i)12-s + 1.21e10·13-s − 1.78e10i·14-s + (−1.51e10 − 5.68e10i)15-s + 1.71e10·16-s − 1.49e11i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.966 + 0.257i)3-s − 0.500·4-s + 1.53i·5-s + (−0.182 − 0.683i)6-s − 1.22·7-s − 0.353i·8-s + (0.867 − 0.497i)9-s − 1.08·10-s − 0.248i·11-s + (0.483 − 0.128i)12-s + 1.14·13-s − 0.863i·14-s + (−0.394 − 1.47i)15-s + 0.250·16-s − 1.26i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.00457753 - 0.00351748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00457753 - 0.00351748i\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 362. iT \) |
| 3 | \( 1 + (1.90e4 - 5.06e3i)T \) |
good | 5 | \( 1 - 2.98e6iT - 3.81e12T^{2} \) |
| 7 | \( 1 + 4.92e7T + 1.62e15T^{2} \) |
| 11 | \( 1 + 5.85e8iT - 5.55e18T^{2} \) |
| 13 | \( 1 - 1.21e10T + 1.12e20T^{2} \) |
| 17 | \( 1 + 1.49e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 3.13e11T + 1.04e23T^{2} \) |
| 23 | \( 1 - 2.41e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 + 1.39e13iT - 2.10e26T^{2} \) |
| 31 | \( 1 + 2.16e13T + 6.99e26T^{2} \) |
| 37 | \( 1 - 2.89e13T + 1.68e28T^{2} \) |
| 41 | \( 1 - 2.60e14iT - 1.07e29T^{2} \) |
| 43 | \( 1 + 9.12e13T + 2.52e29T^{2} \) |
| 47 | \( 1 + 1.34e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 + 4.92e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + 1.63e15iT - 7.50e31T^{2} \) |
| 61 | \( 1 + 1.27e16T + 1.36e32T^{2} \) |
| 67 | \( 1 - 1.62e16T + 7.40e32T^{2} \) |
| 71 | \( 1 + 3.19e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 6.00e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + 1.80e17T + 1.43e34T^{2} \) |
| 83 | \( 1 - 8.49e16iT - 3.49e34T^{2} \) |
| 89 | \( 1 - 5.84e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + 1.70e17T + 5.77e35T^{2} \) |
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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
−18.00726760197210690296629671897, −16.34037577570337190945402210503, −15.28002393663555026796355039246, −13.41706102605803721078177204396, −11.24425454621489668856055647558, −9.793862659824189282330190637081, −6.95601463872881762348848010645, −6.00557962546592460698305099099, −3.51252611718057511956965275021, −0.00306737089505830738251974423,
1.36447798711501239219946032905, 4.24984511827496006736782550584, 6.04050552282655351767962832158, 8.808047610760058026605869492421, 10.57751384456839864342271764464, 12.56126893844413146007477848754, 12.93698323774024478049552779189, 16.11054790008726661032457360954, 17.09706261204694073565059913925, 18.81165512256499529327187625592