| L(s) = 1 | + (−4.13 − 3.00i)2-s + (−1.02 − 9.78i)3-s + (5.61 + 17.2i)4-s + (−1.20 − 2.08i)5-s + (−25.1 + 43.5i)6-s + (9.84 + 2.09i)7-s + (16.0 − 49.4i)8-s + (−68.1 + 14.4i)9-s + (−1.29 + 12.2i)10-s + (19.7 − 21.9i)11-s + (163. − 72.6i)12-s + (4.56 + 2.03i)13-s + (−34.4 − 38.2i)14-s + (−19.1 + 13.9i)15-s + (−97.4 + 70.8i)16-s + (−55.2 − 61.3i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 − 1.06i)2-s + (−0.197 − 1.88i)3-s + (0.701 + 2.15i)4-s + (−0.107 − 0.186i)5-s + (−1.71 + 2.96i)6-s + (0.531 + 0.113i)7-s + (0.709 − 2.18i)8-s + (−2.52 + 0.536i)9-s + (−0.0407 + 0.388i)10-s + (0.540 − 0.600i)11-s + (3.92 − 1.74i)12-s + (0.0974 + 0.0434i)13-s + (−0.657 − 0.730i)14-s + (−0.330 + 0.240i)15-s + (−1.52 + 1.10i)16-s + (−0.788 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.127817 + 0.474451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.127817 + 0.474451i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-149. + 86.0i)T \) |
| good | 2 | \( 1 + (4.13 + 3.00i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (1.02 + 9.78i)T + (-26.4 + 5.61i)T^{2} \) |
| 5 | \( 1 + (1.20 + 2.08i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-9.84 - 2.09i)T + (313. + 139. i)T^{2} \) |
| 11 | \( 1 + (-19.7 + 21.9i)T + (-139. - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (-4.56 - 2.03i)T + (1.47e3 + 1.63e3i)T^{2} \) |
| 17 | \( 1 + (55.2 + 61.3i)T + (-513. + 4.88e3i)T^{2} \) |
| 19 | \( 1 + (74.0 - 32.9i)T + (4.58e3 - 5.09e3i)T^{2} \) |
| 23 | \( 1 + (-4.84 + 14.9i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-110. - 79.9i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 37 | \( 1 + (-80.6 + 139. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-9.01 + 85.7i)T + (-6.74e4 - 1.43e4i)T^{2} \) |
| 43 | \( 1 + (-117. + 52.1i)T + (5.32e4 - 5.90e4i)T^{2} \) |
| 47 | \( 1 + (-232. + 169. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (340. - 72.3i)T + (1.36e5 - 6.05e4i)T^{2} \) |
| 59 | \( 1 + (-0.969 - 9.21i)T + (-2.00e5 + 4.27e4i)T^{2} \) |
| 61 | \( 1 - 454.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-127. - 220. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (501. - 106. i)T + (3.26e5 - 1.45e5i)T^{2} \) |
| 73 | \( 1 + (-695. + 772. i)T + (-4.06e4 - 3.86e5i)T^{2} \) |
| 79 | \( 1 + (-709. - 787. i)T + (-5.15e4 + 4.90e5i)T^{2} \) |
| 83 | \( 1 + (74.0 - 704. i)T + (-5.59e5 - 1.18e5i)T^{2} \) |
| 89 | \( 1 + (-242. - 745. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-129. - 398. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.55729165988444233354779273115, −14.00141076837987966716633298591, −12.66253946462264903238178979232, −11.81090507069115956459712269892, −10.95162750906302119070120069032, −8.823262502979035917950166575288, −8.044277349044019618111245460393, −6.66709414159039279872771584109, −2.32892564261969394801523078005, −0.76139343586835917913497551418,
4.62868340026682789224756438222, 6.35605594134524240391961671913, 8.344513281109748964172682583532, 9.323371991896138481084667709918, 10.39679543519248205840563188918, 11.21952697932312659791703777264, 14.55000481194826622429608391781, 15.19229281049935252948102188110, 15.96266615091049481052844747289, 17.28089904668238367744127003260
