| L(s) = 1 | + (−1.91 + 0.563i)2-s + (5.41 + 6.25i)3-s + (3.36 − 2.16i)4-s + (0.0474 + 0.329i)5-s + (−13.9 − 8.94i)6-s + (−3.79 + 8.31i)7-s + (−5.23 + 6.04i)8-s + (−5.89 + 40.9i)9-s + (−0.276 − 0.606i)10-s + (14.8 + 4.36i)11-s + (31.7 + 9.32i)12-s + (13.9 + 30.4i)13-s + (2.60 − 18.1i)14-s + (−1.80 + 2.08i)15-s + (6.64 − 14.5i)16-s + (−93.2 − 59.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (1.04 + 1.20i)3-s + (0.420 − 0.270i)4-s + (0.00424 + 0.0295i)5-s + (−0.946 − 0.608i)6-s + (−0.205 + 0.449i)7-s + (−0.231 + 0.267i)8-s + (−0.218 + 1.51i)9-s + (−0.00875 − 0.0191i)10-s + (0.407 + 0.119i)11-s + (0.763 + 0.224i)12-s + (0.296 + 0.649i)13-s + (0.0496 − 0.345i)14-s + (−0.0310 + 0.0358i)15-s + (0.103 − 0.227i)16-s + (−1.33 − 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0759 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0759 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.963828 + 0.893203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.963828 + 0.893203i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.91 - 0.563i)T \) |
| 23 | \( 1 + (40.4 + 102. i)T \) |
| good | 3 | \( 1 + (-5.41 - 6.25i)T + (-3.84 + 26.7i)T^{2} \) |
| 5 | \( 1 + (-0.0474 - 0.329i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (3.79 - 8.31i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (-14.8 - 4.36i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (-13.9 - 30.4i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (93.2 + 59.9i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-128. + 82.7i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-139. - 89.7i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 1.18i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (10.9 - 76.2i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (51.4 + 357. i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (142. + 165. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 + 499.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (116. - 255. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (-144. - 315. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (-148. + 171. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (122. - 35.8i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (857. - 251. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (770. - 495. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (99.3 + 217. i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-196. + 1.36e3i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-182. - 210. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-115. - 806. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−15.81533523041639068527839636058, −14.64735789285168341840319696057, −13.70487555439519891517195245689, −11.70423082257322762247564847627, −10.38491434258889146124379297663, −9.173483128984765772989647056292, −8.748488186205425421340507589480, −6.89391540554817792044884561947, −4.69951273358941643822948627381, −2.80887443720123202551382523980,
1.41214322954360248127744207437, 3.28145448477551505286356036679, 6.48923334147936526982503744519, 7.70340281182133910052810121686, 8.597668834747301604191708110280, 9.933133816169306383504002941296, 11.54461974242770599697945603457, 12.86179888654577326225089576877, 13.65075232985845555907535170860, 14.86354329071490025039428571635
