| L(s) = 1 | + (−3.23 + 2.35i)2-s + (−4.88 + 15.0i)3-s + (4.94 − 15.2i)4-s + (51.0 − 22.8i)5-s + (−19.5 − 60.1i)6-s + 188.·7-s + (19.7 + 60.8i)8-s + (−5.97 − 4.34i)9-s + (−111. + 193. i)10-s + (−312. + 226. i)11-s + (204. + 148. i)12-s + (−145. − 105. i)13-s + (−610. + 443. i)14-s + (95.0 + 879. i)15-s + (−207. − 150. i)16-s + (199. + 614. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.313 + 0.965i)3-s + (0.154 − 0.475i)4-s + (0.912 − 0.409i)5-s + (−0.221 − 0.682i)6-s + 1.45·7-s + (0.109 + 0.336i)8-s + (−0.0245 − 0.0178i)9-s + (−0.351 + 0.613i)10-s + (−0.778 + 0.565i)11-s + (0.410 + 0.298i)12-s + (−0.238 − 0.173i)13-s + (−0.832 + 0.605i)14-s + (0.109 + 1.00i)15-s + (−0.202 − 0.146i)16-s + (0.167 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0269 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0269 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.01605 + 1.04377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01605 + 1.04377i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 5 | \( 1 + (-51.0 + 22.8i)T \) |
| good | 3 | \( 1 + (4.88 - 15.0i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 - 188.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (312. - 226. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (145. + 105. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-199. - 614. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-632. - 1.94e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.86e3 + 1.35e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (584. - 1.80e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.34e3 - 4.12e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (6.10e3 + 4.43e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.03e4 + 7.52e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 2.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-595. + 1.83e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (7.33e3 - 2.25e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (22.8 + 16.5i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-7.81e3 + 5.68e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.02e4 + 3.16e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.23e4 - 6.87e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.89e4 + 4.28e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-3.17e4 + 9.75e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.52e4 + 1.08e5i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-3.84e4 + 2.79e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.16e3 - 1.28e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.98778527617060315873790122547, −14.08817728229514456372897772449, −12.45387897538545051150586773239, −10.73562277625599562822970931992, −10.19979269457105751745095752689, −8.875289381626350502538980188715, −7.61003447752766587498773870795, −5.54328035133148496520379488423, −4.74637883037887551860448783406, −1.69356481163227131498057491003,
1.06510016209025785704161744180, 2.42985378773494010968994275978, 5.25162984179126397104964388313, 6.91514010014323764154242575412, 7.980546944215026634701803330909, 9.481031895866438465873239059591, 10.92899845241948995911938091518, 11.67354384466974855167549688014, 13.12660877461599631106795149183, 13.91614412790274973078369934549
