| L(s) = 1 | + (−3.24 − 2.03i)2-s + (−2.23 + 0.780i)3-s + (4.63 + 9.61i)4-s + (−5.12 + 1.17i)5-s + (8.82 + 2.01i)6-s + (−6.56 − 3.16i)7-s + (2.86 − 25.4i)8-s + (−2.67 + 2.13i)9-s + (19.0 + 6.65i)10-s + (−0.480 − 4.26i)11-s + (−17.8 − 17.8i)12-s + (2.73 + 2.17i)13-s + (14.8 + 23.6i)14-s + (10.5 − 6.61i)15-s + (−34.4 + 43.2i)16-s + (−6.15 + 6.15i)17-s + ⋯ |
| L(s) = 1 | + (−1.62 − 1.01i)2-s + (−0.743 + 0.260i)3-s + (1.15 + 2.40i)4-s + (−1.02 + 0.234i)5-s + (1.47 + 0.335i)6-s + (−0.938 − 0.451i)7-s + (0.357 − 3.17i)8-s + (−0.296 + 0.236i)9-s + (1.90 + 0.665i)10-s + (−0.0437 − 0.387i)11-s + (−1.48 − 1.48i)12-s + (0.210 + 0.167i)13-s + (1.06 + 1.68i)14-s + (0.701 − 0.440i)15-s + (−2.15 + 2.70i)16-s + (−0.362 + 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00916210 + 0.0210649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00916210 + 0.0210649i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (0.661 - 28.9i)T \) |
| good | 2 | \( 1 + (3.24 + 2.03i)T + (1.73 + 3.60i)T^{2} \) |
| 3 | \( 1 + (2.23 - 0.780i)T + (7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (5.12 - 1.17i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 + (6.56 + 3.16i)T + (30.5 + 38.3i)T^{2} \) |
| 11 | \( 1 + (0.480 + 4.26i)T + (-117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-2.73 - 2.17i)T + (37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (6.15 - 6.15i)T - 289iT^{2} \) |
| 19 | \( 1 + (-5.18 + 14.8i)T + (-282. - 225. i)T^{2} \) |
| 23 | \( 1 + (3.58 - 15.7i)T + (-476. - 229. i)T^{2} \) |
| 31 | \( 1 + (34.6 + 21.8i)T + (416. + 865. i)T^{2} \) |
| 37 | \( 1 + (5.62 - 49.9i)T + (-1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (0.940 + 0.940i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (10.2 + 16.3i)T + (-802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (59.1 - 6.66i)T + (2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (8.89 + 38.9i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 - 6.74T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-74.0 + 25.9i)T + (2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (-29.6 + 23.6i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-65.3 - 52.0i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (38.0 - 23.8i)T + (2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (120. + 13.6i)T + (6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (-7.07 + 3.40i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (94.3 + 59.3i)T + (3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (13.0 + 4.56i)T + (7.35e3 + 5.86e3i)T^{2} \) |
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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
−17.42744572115861841452773928672, −16.48315914365042584666768014251, −15.78458805813984800796246179087, −13.03513826772927490015549486029, −11.57638043757604854141600071158, −11.07048394703905421537377739180, −9.845489925486203494800165633379, −8.372370253764244689510847521424, −6.95912847978972738639899830936, −3.46634478136846994510818229595,
0.05084524685827240654317412974, 5.80355889069658188716128464523, 7.00249672070072389800738930879, 8.358859343762755575316241919745, 9.612823707609008283586474559180, 11.09351881550986896700829669709, 12.29813590574804372729865809783, 14.73763408081321576333734425535, 15.91168729542790061802049600618, 16.39893189425842696473278772728
