| L(s) = 1 | + (0.830 − 1.81i)2-s + (−5.88 − 1.72i)3-s + (−2.61 − 3.02i)4-s + (−11.4 + 7.33i)5-s + (−8.03 + 9.27i)6-s + (−0.0110 + 0.0771i)7-s + (−7.67 + 2.25i)8-s + (8.95 + 5.75i)9-s + (3.85 + 26.8i)10-s + (−18.6 − 40.8i)11-s + (10.1 + 22.3i)12-s + (−3.95 − 27.5i)13-s + (0.131 + 0.0842i)14-s + (79.8 − 23.4i)15-s + (−2.27 + 15.8i)16-s + (−11.8 + 13.6i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−1.13 − 0.332i)3-s + (−0.327 − 0.377i)4-s + (−1.02 + 0.655i)5-s + (−0.546 + 0.631i)6-s + (−0.000598 + 0.00416i)7-s + (−0.339 + 0.0996i)8-s + (0.331 + 0.213i)9-s + (0.122 + 0.848i)10-s + (−0.510 − 1.11i)11-s + (0.245 + 0.537i)12-s + (−0.0844 − 0.587i)13-s + (0.00250 + 0.00160i)14-s + (1.37 − 0.403i)15-s + (−0.0355 + 0.247i)16-s + (−0.169 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0423408 + 0.311505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0423408 + 0.311505i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.830 + 1.81i)T \) |
| 23 | \( 1 + (-16.7 + 109. i)T \) |
| good | 3 | \( 1 + (5.88 + 1.72i)T + (22.7 + 14.5i)T^{2} \) |
| 5 | \( 1 + (11.4 - 7.33i)T + (51.9 - 113. i)T^{2} \) |
| 7 | \( 1 + (0.0110 - 0.0771i)T + (-329. - 96.6i)T^{2} \) |
| 11 | \( 1 + (18.6 + 40.8i)T + (-871. + 1.00e3i)T^{2} \) |
| 13 | \( 1 + (3.95 + 27.5i)T + (-2.10e3 + 618. i)T^{2} \) |
| 17 | \( 1 + (11.8 - 13.6i)T + (-699. - 4.86e3i)T^{2} \) |
| 19 | \( 1 + (0.965 + 1.11i)T + (-976. + 6.78e3i)T^{2} \) |
| 29 | \( 1 + (178. - 205. i)T + (-3.47e3 - 2.41e4i)T^{2} \) |
| 31 | \( 1 + (-294. + 86.5i)T + (2.50e4 - 1.61e4i)T^{2} \) |
| 37 | \( 1 + (319. + 205. i)T + (2.10e4 + 4.60e4i)T^{2} \) |
| 41 | \( 1 + (251. - 161. i)T + (2.86e4 - 6.26e4i)T^{2} \) |
| 43 | \( 1 + (191. + 56.2i)T + (6.68e4 + 4.29e4i)T^{2} \) |
| 47 | \( 1 + 380.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-78.7 + 547. i)T + (-1.42e5 - 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-17.6 - 122. i)T + (-1.97e5 + 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-375. + 110. i)T + (1.90e5 - 1.22e5i)T^{2} \) |
| 67 | \( 1 + (-237. + 520. i)T + (-1.96e5 - 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-255. + 558. i)T + (-2.34e5 - 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-158. - 182. i)T + (-5.53e4 + 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-59.1 - 411. i)T + (-4.73e5 + 1.38e5i)T^{2} \) |
| 83 | \( 1 + (949. + 609. i)T + (2.37e5 + 5.20e5i)T^{2} \) |
| 89 | \( 1 + (121. + 35.6i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-280. + 180. i)T + (3.79e5 - 8.30e5i)T^{2} \) |
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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
−14.63716867916189783529921586463, −13.16061834833006269253416223804, −12.05301383115094436288975617600, −11.19168829486435188875229411714, −10.51618081524926063715477960432, −8.340811855404222498481617135703, −6.69526716888009948836978514234, −5.28427206114706819509967698644, −3.33786341467040659460211240038, −0.25196409826475903033555665905,
4.31533550126288989933263240591, 5.27159965751773065279910045365, 6.94292126041363673443238159751, 8.271757549667378074803742408855, 9.951225357655008018396142321079, 11.59449071943598851069830320515, 12.17292636075632613031303232225, 13.55344127870060999447454391800, 15.30929079131707049458664767522, 15.80119056768691766173276800577
