| L(s) = 1 | + (−2.68 − 1.72i)2-s + (−0.220 − 1.53i)3-s + (0.905 + 1.98i)4-s + (−18.5 − 5.43i)5-s + (−2.05 + 4.49i)6-s + (13.0 − 15.0i)7-s + (−2.64 + 18.3i)8-s + (23.5 − 6.92i)9-s + (40.3 + 46.5i)10-s + (14.4 − 9.31i)11-s + (2.84 − 1.82i)12-s + (−24.7 − 28.5i)13-s + (−60.7 + 17.8i)14-s + (−4.25 + 29.6i)15-s + (50.2 − 57.9i)16-s + (−9.20 + 20.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.948 − 0.609i)2-s + (−0.0424 − 0.295i)3-s + (0.113 + 0.247i)4-s + (−1.65 − 0.486i)5-s + (−0.139 + 0.306i)6-s + (0.702 − 0.810i)7-s + (−0.116 + 0.812i)8-s + (0.874 − 0.256i)9-s + (1.27 + 1.47i)10-s + (0.397 − 0.255i)11-s + (0.0683 − 0.0439i)12-s + (−0.527 − 0.608i)13-s + (−1.16 + 0.340i)14-s + (−0.0733 + 0.509i)15-s + (0.784 − 0.905i)16-s + (−0.131 + 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.157842 - 0.488467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157842 - 0.488467i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (73.3 + 82.3i)T \) |
| good | 2 | \( 1 + (2.68 + 1.72i)T + (3.32 + 7.27i)T^{2} \) |
| 3 | \( 1 + (0.220 + 1.53i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (18.5 + 5.43i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-13.0 + 15.0i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-14.4 + 9.31i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (24.7 + 28.5i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (9.20 - 20.1i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-2.80 - 6.13i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-94.1 + 206. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (41.6 - 289. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-109. + 32.0i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-425. - 124. i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (39.5 + 274. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 75.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-128. + 148. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-232. - 267. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (5.14 - 35.7i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (33.9 + 21.8i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-534. - 343. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-147. - 322. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (475. + 548. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (1.14e3 - 335. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (230. + 1.60e3i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (595. + 174. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−17.18163959773732727091714791772, −15.83265640744765158054358426423, −14.46038280011177497032028881655, −12.49087612673229788967070392315, −11.46393112029496120359024960364, −10.26188189223725261434387869914, −8.455406029703596865558760069946, −7.50572442448797094252615829594, −4.33624154639753446641721469875, −0.802558627922171910983636632617,
4.21785029821062194352056399757, 7.12306563043988467364124136998, 8.043172567406993662842833113353, 9.484185957603435567820871730259, 11.25132454503483350340986445798, 12.37649612940771471374282960233, 14.78351849235580903052248400002, 15.62180454439704205934513238703, 16.43881257171338866277122187072, 18.00986862484500107218921207873
