| L(s) = 1 | + (1.06e4 + 1.84e4i)2-s + (2.39e6 − 4.14e6i)3-s + (4.11e7 − 7.12e7i)4-s + (−5.16e9 − 8.95e9i)5-s + (1.01e11 + 7.62e−6i)6-s + (−1.66e12 + 6.66e11i)7-s + (1.32e13 + 0.000244i)8-s + (−1.14e13 − 1.98e13i)9-s + (1.10e14 − 1.90e14i)10-s + (2.52e14 − 4.37e14i)11-s + (−1.96e14 − 3.40e14i)12-s − 9.24e15·13-s + (−3.00e16 − 2.36e16i)14-s − 4.94e16·15-s + (1.18e17 + 2.05e17i)16-s + (9.70e16 − 1.68e17i)17-s + ⋯ |
| L(s) = 1 | + (0.460 + 0.796i)2-s + (0.288 − 0.499i)3-s + (0.0765 − 0.132i)4-s + (−0.378 − 0.656i)5-s + 0.531·6-s + (−0.928 + 0.371i)7-s + 1.06·8-s + (−0.166 − 0.288i)9-s + (0.348 − 0.603i)10-s + (0.200 − 0.347i)11-s + (−0.0442 − 0.0765i)12-s − 0.651·13-s + (−0.723 − 0.568i)14-s − 0.437·15-s + (0.411 + 0.713i)16-s + (0.139 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.09434949546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09434949546\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 + 4.14e6i)T \) |
| 7 | \( 1 + (1.66e12 - 6.66e11i)T \) |
| good | 2 | \( 1 + (-1.06e4 - 1.84e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (5.16e9 + 8.95e9i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-2.52e14 + 4.37e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 9.24e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-9.70e16 + 1.68e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-6.43e17 - 1.11e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (1.91e19 + 3.31e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 6.81e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-6.77e20 + 1.17e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (8.26e21 + 1.43e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 + 1.71e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 2.80e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (1.87e23 + 3.24e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (7.52e24 - 1.30e25i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (4.19e25 - 7.26e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-4.00e25 - 6.93e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.38e26 - 2.39e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 3.50e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (8.21e26 - 1.42e27i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (-1.82e27 - 3.15e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 7.43e26T + 4.50e55T^{2} \) |
| 89 | \( 1 + (6.92e27 + 1.19e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 9.55e27T + 4.13e57T^{2} \) |
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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
−11.89674935381303009501886741717, −10.12548418795193484367450314282, −8.772531321290675702090232372359, −7.54643219689884977648294012887, −6.48358260734906109782869688557, −5.47958329529894616066764469589, −4.17050435707042125107599964016, −2.66371457859472239305866330953, −1.22589525124729183961680988755, −0.01547130952884133792071504284,
1.80078725210352462854450740128, 3.09765217087466734495121803362, 3.58307770610027694337646258892, 4.83037110433615174004558871733, 6.74455867558710936670186185349, 7.75173307523362852943993858918, 9.558350312572496701598986603500, 10.49601273521651789638787547067, 11.56627900687559066844578458225, 12.69834271160160788464621071428
