| L(s) = 1 | + (−0.540 + 2.77i)2-s + (2.23 − 2.23i)3-s + (−7.41 − 3.00i)4-s + (5 + 7.41i)6-s + (11.1 + 11.1i)7-s + (12.3 − 18.9i)8-s + 17i·9-s + 59.3i·11-s + (−23.2 + 9.87i)12-s + (26.5 + 26.5i)13-s + (−37.0 + 25.0i)14-s + (46.0 + 44.4i)16-s + (79.5 − 79.5i)17-s + (−47.1 − 9.18i)18-s − 59.3·19-s + ⋯ |
| L(s) = 1 | + (−0.191 + 0.981i)2-s + (0.430 − 0.430i)3-s + (−0.927 − 0.375i)4-s + (0.340 + 0.504i)6-s + (0.603 + 0.603i)7-s + (0.545 − 0.838i)8-s + 0.629i·9-s + 1.62i·11-s + (−0.560 + 0.237i)12-s + (0.566 + 0.566i)13-s + (−0.707 + 0.477i)14-s + (0.718 + 0.695i)16-s + (1.13 − 1.13i)17-s + (−0.618 − 0.120i)18-s − 0.716·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.977108 + 1.15815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.977108 + 1.15815i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 - 2.77i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-2.23 + 2.23i)T - 27iT^{2} \) |
| 7 | \( 1 + (-11.1 - 11.1i)T + 343iT^{2} \) |
| 11 | \( 1 - 59.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-26.5 - 26.5i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + (-79.5 + 79.5i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 59.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (105. - 105. i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + 46iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 118. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-212. + 212. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 188T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-122. + 122. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-194. - 194. i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-26.5 - 26.5i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 72T + 2.26e5T^{2} \) |
| 67 | \( 1 + (597. + 597. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 711. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (504. + 504. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-221. + 221. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 726iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (451. - 451. i)T - 9.12e5iT^{2} \) |
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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
−13.95246056938039224767463055004, −12.90730574156608522341410658129, −11.74066220098398939460728436813, −10.09469054081926121279381015004, −9.079920247762124331197061254075, −7.890017221075644253930885514365, −7.23704498377957265158464125588, −5.65853150826995558753584228259, −4.44073936270969215907581987365, −1.91382699516736112696622837417,
0.993588620228450646734464364849, 3.20029276375015889396569524863, 4.14360729884278268933589301396, 5.94060439946754719806905488148, 8.180667368442799900074793165040, 8.626485968960231181799443134205, 10.16440880190914111181724844874, 10.75875022583717690534727173288, 11.90797162420454781005686265404, 13.04200166000525778512978101765
