L(s) = 1 | + (−1.37 + 0.663i)2-s + (−4.45 − 5.58i)3-s + (−18.4 + 23.1i)4-s + (71.8 − 34.6i)5-s + (9.84 + 4.74i)6-s + (−153. − 191. i)7-s + (20.9 − 91.8i)8-s + (42.7 − 187. i)9-s + (−76.0 + 95.3i)10-s + (22.8 + 100. i)11-s + 212.·12-s + (−70.1 − 307. i)13-s + (337. + 162. i)14-s + (−513. − 247. i)15-s + (−179. − 784. i)16-s + 1.22e3·17-s + ⋯ |
L(s) = 1 | + (−0.243 + 0.117i)2-s + (−0.285 − 0.358i)3-s + (−0.577 + 0.724i)4-s + (1.28 − 0.619i)5-s + (0.111 + 0.0537i)6-s + (−1.18 − 1.48i)7-s + (0.115 − 0.507i)8-s + (0.175 − 0.769i)9-s + (−0.240 + 0.301i)10-s + (0.0570 + 0.249i)11-s + 0.425·12-s + (−0.115 − 0.504i)13-s + (0.460 + 0.221i)14-s + (−0.589 − 0.283i)15-s + (−0.174 − 0.766i)16-s + 1.02·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0428 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0428 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.680950 - 0.710811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.680950 - 0.710811i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (2.17e3 + 3.97e3i)T \) |
good | 2 | \( 1 + (1.37 - 0.663i)T + (19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (4.45 + 5.58i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-71.8 + 34.6i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + (153. + 191. i)T + (-3.73e3 + 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-22.8 - 100. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (70.1 + 307. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 - 1.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (1.32e3 - 1.66e3i)T + (-5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (-355. - 171. i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 31 | \( 1 + (928. - 447. i)T + (1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (-1.64e3 + 7.20e3i)T + (-6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 - 8.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-2.09e4 - 1.00e4i)T + (9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (3.57e3 + 1.56e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (9.63e3 - 4.63e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 - 2.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (7.00e3 + 8.77e3i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-4.20e3 + 1.84e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-1.36e4 - 5.99e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-2.70e4 - 1.30e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (-1.40e4 + 6.15e4i)T + (-2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (-2.04e4 + 2.56e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.02e4 + 2.89e4i)T + (3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-3.17e4 + 3.98e4i)T + (-1.91e9 - 8.37e9i)T^{2} \) |
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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
−16.38976005392689545782911076471, −14.16521424589391068698982108164, −12.95959405730095796379084014048, −12.64444035957647231215186410948, −10.10304781470097704904791887837, −9.406613533613007276136345872278, −7.51439406228865409407297715525, −6.07573034677351707129347349554, −3.83779775213612640601754438219, −0.72602780837050622823510042902,
2.35592505217017940311026917638, 5.33670995806594985527249978484, 6.27178561426189754984285518418, 9.066547832710319943967853630234, 9.801146985184957955754666866140, 10.89664335448975596154329693099, 12.82965896644859078838903992825, 13.96620518252244417366027692760, 15.12041547387883688572798511481, 16.44345350937727568654419311405