L(s) = 1 | + (1.40 + 0.187i)2-s + (−2.02 + 0.794i)3-s + (1.93 + 0.524i)4-s + (−0.100 + 0.663i)5-s + (−2.98 + 0.735i)6-s + (−1.93 + 1.80i)7-s + (2.60 + 1.09i)8-s + (1.26 − 1.17i)9-s + (−0.264 + 0.911i)10-s + (−1.41 + 1.52i)11-s + (−4.32 + 0.472i)12-s + (−0.994 − 0.226i)13-s + (−3.05 + 2.16i)14-s + (−0.324 − 1.42i)15-s + (3.45 + 2.02i)16-s + (−0.148 + 1.98i)17-s + ⋯ |
L(s) = 1 | + (0.991 + 0.132i)2-s + (−1.16 + 0.458i)3-s + (0.965 + 0.262i)4-s + (−0.0447 + 0.296i)5-s + (−1.21 + 0.300i)6-s + (−0.732 + 0.680i)7-s + (0.921 + 0.387i)8-s + (0.422 − 0.391i)9-s + (−0.0835 + 0.288i)10-s + (−0.426 + 0.460i)11-s + (−1.24 + 0.136i)12-s + (−0.275 − 0.0629i)13-s + (−0.816 + 0.577i)14-s + (−0.0838 − 0.367i)15-s + (0.862 + 0.505i)16-s + (−0.0360 + 0.481i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.673198 + 1.17826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.673198 + 1.17826i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.187i)T \) |
| 7 | \( 1 + (1.93 - 1.80i)T \) |
good | 3 | \( 1 + (2.02 - 0.794i)T + (2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (0.100 - 0.663i)T + (-4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (1.41 - 1.52i)T + (-0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.994 + 0.226i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.148 - 1.98i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (2.77 - 1.60i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.114 - 1.52i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (-0.236 - 0.490i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (3.82 - 6.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.55 + 5.21i)T + (-13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (-1.99 + 2.49i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-3.95 + 3.15i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.00 + 2.46i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (-0.0937 - 0.137i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-0.124 - 0.824i)T + (-56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (1.97 - 2.89i)T + (-22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 5.80i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (12.2 + 5.87i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-8.28 - 2.55i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (3.13 + 5.42i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-14.2 + 3.26i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-9.30 + 8.63i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 - 1.06T + 97T^{2} \) |
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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
−11.77091995558294019800627342516, −10.70455799280769589967541301791, −10.36368024925201053968603448913, −8.938355042253844611132558619297, −7.47541203591145493940304562477, −6.50653053340568331615852958387, −5.71445018781516519813011190468, −4.99192504228486322309083078884, −3.78753303997723680623842874472, −2.43543251845642277405786302819,
0.73506552206197929991540345419, 2.75168257287915590724534628848, 4.20215586749482917923879920779, 5.18975958461905772554994326216, 6.17821858445943641395496696271, 6.78845265310875693613331865331, 7.79409657521521824656311061716, 9.425877897065966803070244404175, 10.62255549006579104367084949705, 11.09950344761417265962364907699