| L(s) = 1 | + (0.104 + 0.994i)2-s + (−1.32 − 1.47i)3-s + (−0.978 + 0.207i)4-s + (1.09 + 1.94i)5-s + (1.32 − 1.47i)6-s − 0.0211·7-s + (−0.309 − 0.951i)8-s + (−0.0987 + 0.939i)9-s + (−1.82 + 1.29i)10-s + (−0.534 + 0.388i)11-s + (1.60 + 1.16i)12-s + (0.501 − 4.77i)13-s + (−0.00221 − 0.0210i)14-s + (1.42 − 4.20i)15-s + (0.913 − 0.406i)16-s + (0.491 + 0.104i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.767 − 0.852i)3-s + (−0.489 + 0.103i)4-s + (0.490 + 0.871i)5-s + (0.542 − 0.602i)6-s − 0.00800·7-s + (−0.109 − 0.336i)8-s + (−0.0329 + 0.313i)9-s + (−0.576 + 0.409i)10-s + (−0.161 + 0.117i)11-s + (0.463 + 0.337i)12-s + (0.139 − 1.32i)13-s + (−0.000591 − 0.00562i)14-s + (0.366 − 1.08i)15-s + (0.228 − 0.101i)16-s + (0.119 + 0.0253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24359 - 0.0571913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24359 - 0.0571913i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-1.09 - 1.94i)T \) |
| 19 | \( 1 + (-4.00 - 1.72i)T \) |
| good | 3 | \( 1 + (1.32 + 1.47i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 0.0211T + 7T^{2} \) |
| 11 | \( 1 + (0.534 - 0.388i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.501 + 4.77i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.491 - 0.104i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (4.14 + 1.84i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-7.04 + 1.49i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (0.850 + 2.61i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.78 - 4.20i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-10.8 + 4.80i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-4.34 + 7.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-11.1 + 2.36i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-1.74 + 0.370i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (1.46 - 0.650i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-1.98 - 0.882i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-5.36 + 5.95i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-0.526 - 0.584i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (1.59 + 15.1i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.998 - 1.10i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (1.71 + 5.26i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-12.9 - 5.75i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-0.451 - 0.501i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.14561352810865064851519246259, −9.187076166905019987614061358864, −7.81646408372340091351891575330, −7.53042892461421242922715708501, −6.41285901048548231870875795463, −5.98304708962553094105031520923, −5.24270761827877984326793608679, −3.73215135065117319056879737306, −2.49110923290932282233172767118, −0.790314186110806532129361177671,
1.11409411042016966182576369185, 2.51025797125834783352515325712, 4.09331862512072888042016765211, 4.61141328038812720220072106239, 5.49281680078412756664892659331, 6.22893343530711745951696971759, 7.67681866444886552553156301235, 8.771143478538469516110707989045, 9.556277666957819038959161460102, 9.924498855022007648207132344308
