| L(s) = 1 | + (1.68 + 1.08i)2-s + (1.01 + 7.04i)3-s + (1.66 + 3.63i)4-s + (−7.67 − 2.25i)5-s + (−5.91 + 12.9i)6-s + (4.19 − 4.84i)7-s + (−1.13 + 7.91i)8-s + (−22.7 + 6.66i)9-s + (−10.4 − 12.0i)10-s + (37.3 − 23.9i)11-s + (−23.9 + 15.3i)12-s + (49.1 + 56.7i)13-s + (12.3 − 3.61i)14-s + (8.10 − 56.3i)15-s + (−10.4 + 12.0i)16-s + (14.7 − 32.3i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.194 + 1.35i)3-s + (0.207 + 0.454i)4-s + (−0.686 − 0.201i)5-s + (−0.402 + 0.881i)6-s + (0.226 − 0.261i)7-s + (−0.0503 + 0.349i)8-s + (−0.840 + 0.246i)9-s + (−0.331 − 0.382i)10-s + (1.02 − 0.657i)11-s + (−0.576 + 0.370i)12-s + (1.04 + 1.21i)13-s + (0.234 − 0.0689i)14-s + (0.139 − 0.969i)15-s + (−0.163 + 0.188i)16-s + (0.210 − 0.462i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0792 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0792 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.22538 + 1.32673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22538 + 1.32673i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 - 1.08i)T \) |
| 23 | \( 1 + (102. + 41.8i)T \) |
| good | 3 | \( 1 + (-1.01 - 7.04i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (7.67 + 2.25i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-4.19 + 4.84i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-37.3 + 23.9i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-49.1 - 56.7i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-14.7 + 32.3i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (29.3 + 64.2i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-63.7 + 139. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-14.8 + 103. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (363. - 106. i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-259. - 76.2i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-26.0 - 181. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-166. + 192. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (511. + 590. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-43.9 + 305. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-503. - 323. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (649. + 417. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-345. - 756. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (365. + 422. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-546. + 160. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-154. - 1.07e3i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-1.01e3 - 297. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−15.79918154630186259650805353051, −14.50583026715679849589720981599, −13.73085231318126458000388956185, −11.85355822651637529703298455145, −11.05002545025574398157985522397, −9.397083827286958833290178019321, −8.296164428036737929399237524986, −6.41069138303282546684190948400, −4.49838190711789450053596055217, −3.76200754172759190334484274735,
1.57743341605749393653157450743, 3.67685275585860319542249092399, 5.94793926915645255106967498921, 7.28675518474006745667056950282, 8.472667600249053576497100477114, 10.53095455942726593080403474620, 11.97456861728841794115346323785, 12.47851032057319861499479046389, 13.68213926197243996360557937160, 14.70236647749350583939918717751
