| 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
−17.64744012869155402065640923015, −16.24767330597467916631292307699, −15.33276001472637837814388413706, −14.72930996204299335905513814457, −12.94461052582339151269348160920, −11.10374095581219019306309657847, −8.976786216122599869085921042353, −8.180751835384302776233898736263, −6.37010572994865227344422484709, −4.64206805355458542275733214452,
1.77453611027848162143355899667, 4.07998957597926559240161592448, 7.23523781299397609812699072602, 8.954487682455748359110530141333, 10.84998392450067036548611795063, 11.26887112089041118859308276355, 12.97402842612825385167613497248, 13.90379790239724823930991625597, 15.67547849819596343420786583929, 17.50168323000441655589515596379
