| L(s) = 1 | + (−1.41 + 0.0458i)2-s + (−0.204 − 1.71i)3-s + (1.99 − 0.129i)4-s + (−0.561 − 0.300i)5-s + (0.367 + 2.42i)6-s + (−0.946 − 0.188i)7-s + (−2.81 + 0.274i)8-s + (−2.91 + 0.701i)9-s + (0.807 + 0.398i)10-s + (−4.33 + 3.56i)11-s + (−0.630 − 3.40i)12-s + (−1.08 + 0.579i)13-s + (1.34 + 0.222i)14-s + (−0.401 + 1.02i)15-s + (3.96 − 0.517i)16-s + (2.91 − 1.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0324i)2-s + (−0.117 − 0.993i)3-s + (0.997 − 0.0648i)4-s + (−0.251 − 0.134i)5-s + (0.149 + 0.988i)6-s + (−0.357 − 0.0711i)7-s + (−0.995 + 0.0971i)8-s + (−0.972 + 0.233i)9-s + (0.255 + 0.126i)10-s + (−1.30 + 1.07i)11-s + (−0.181 − 0.983i)12-s + (−0.300 + 0.160i)13-s + (0.359 + 0.0594i)14-s + (−0.103 + 0.265i)15-s + (0.991 − 0.129i)16-s + (0.706 − 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0388330 + 0.0727908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0388330 + 0.0727908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0458i)T \) |
| 3 | \( 1 + (0.204 + 1.71i)T \) |
| good | 5 | \( 1 + (0.561 + 0.300i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (0.946 + 0.188i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (4.33 - 3.56i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.08 - 0.579i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-2.91 + 1.20i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.764 - 2.51i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (3.58 + 2.39i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-4.42 - 3.63i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (5.09 - 5.09i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.98 - 9.82i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (6.17 + 4.12i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.337 + 3.42i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (5.09 - 2.11i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 4.26i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (5.41 + 2.89i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-1.20 - 12.2i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (6.84 - 0.673i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (1.54 + 0.306i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (0.975 + 4.90i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.41 + 8.25i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-3.20 + 10.5i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (1.88 + 2.82i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (7.71 + 7.71i)T + 97iT^{2} \) |
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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
−11.86553665705405408891766099423, −10.43501332090764779817665518531, −10.02325591029745982141690889595, −8.642725578226091821012417733357, −7.891644078470937752761771195317, −7.20962340525130893434759737256, −6.29454615347187434896128350760, −5.08889156904186406631989761752, −2.99170668005175547436504503543, −1.80414884748536329420985977710,
0.07068416659578919153926091454, 2.67484345212459616259840855380, 3.65073004858278821836149249075, 5.38553118050548345076998521457, 6.14098490369948519009660464709, 7.67090609582278698837533573414, 8.275491669912757099327228173894, 9.407733234609293844898309112611, 10.01188815440168301552856361315, 10.93137295291006240909871199937
