| L(s) = 1 | + (−0.589 − 4.09i)2-s + (1.03 − 2.27i)3-s + (−8.77 + 2.57i)4-s + (−3.66 + 4.23i)5-s + (−9.94 − 2.91i)6-s + (16.9 − 10.9i)7-s + (1.97 + 4.32i)8-s + (13.5 + 15.6i)9-s + (19.5 + 12.5i)10-s + (0.251 − 1.74i)11-s + (−3.25 + 22.6i)12-s + (53.9 + 34.6i)13-s + (−54.7 − 63.1i)14-s + (5.82 + 12.7i)15-s + (−45.0 + 28.9i)16-s + (−89.5 − 26.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.208 − 1.44i)2-s + (0.200 − 0.438i)3-s + (−1.09 + 0.322i)4-s + (−0.327 + 0.378i)5-s + (−0.676 − 0.198i)6-s + (0.916 − 0.589i)7-s + (0.0872 + 0.190i)8-s + (0.502 + 0.580i)9-s + (0.616 + 0.396i)10-s + (0.00688 − 0.0478i)11-s + (−0.0783 + 0.545i)12-s + (1.15 + 0.740i)13-s + (−1.04 − 1.20i)14-s + (0.100 + 0.219i)15-s + (−0.703 + 0.452i)16-s + (−1.27 − 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.554213 - 0.930766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.554213 - 0.930766i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-69.5 - 85.5i)T \) |
| good | 2 | \( 1 + (0.589 + 4.09i)T + (-7.67 + 2.25i)T^{2} \) |
| 3 | \( 1 + (-1.03 + 2.27i)T + (-17.6 - 20.4i)T^{2} \) |
| 5 | \( 1 + (3.66 - 4.23i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-16.9 + 10.9i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (-0.251 + 1.74i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (-53.9 - 34.6i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (89.5 + 26.2i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (50.7 - 14.8i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (219. + 64.5i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-14.1 - 31.0i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (141. + 162. i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-271. + 313. i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-114. + 250. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 + 100.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (392. - 252. i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-184. - 118. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (215. + 472. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-132. - 924. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (28.3 + 197. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (273. - 80.2i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-186. - 119. i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (390. + 451. i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (179. - 393. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-651. + 752. i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−17.50168323000441655589515596379, −15.67547849819596343420786583929, −13.90379790239724823930991625597, −12.97402842612825385167613497248, −11.26887112089041118859308276355, −10.84998392450067036548611795063, −8.954487682455748359110530141333, −7.23523781299397609812699072602, −4.07998957597926559240161592448, −1.77453611027848162143355899667,
4.64206805355458542275733214452, 6.37010572994865227344422484709, 8.180751835384302776233898736263, 8.976786216122599869085921042353, 11.10374095581219019306309657847, 12.94461052582339151269348160920, 14.72930996204299335905513814457, 15.33276001472637837814388413706, 16.24767330597467916631292307699, 17.64744012869155402065640923015
