| L(s) = 1 | − 0.164·2-s + (1.34 + 1.08i)3-s − 1.97·4-s + i·5-s + (−0.221 − 0.179i)6-s + i·7-s + 0.654·8-s + (0.630 + 2.93i)9-s − 0.164i·10-s + (−3.08 + 1.22i)11-s + (−2.65 − 2.14i)12-s + 1.60i·13-s − 0.164i·14-s + (−1.08 + 1.34i)15-s + 3.83·16-s + 5.06·17-s + ⋯ |
| L(s) = 1 | − 0.116·2-s + (0.777 + 0.628i)3-s − 0.986·4-s + 0.447i·5-s + (−0.0906 − 0.0732i)6-s + 0.377i·7-s + 0.231·8-s + (0.210 + 0.977i)9-s − 0.0520i·10-s + (−0.929 + 0.369i)11-s + (−0.767 − 0.619i)12-s + 0.445i·13-s − 0.0440i·14-s + (−0.281 + 0.347i)15-s + 0.959·16-s + 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9854030902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9854030902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.34 - 1.08i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (3.08 - 1.22i)T \) |
| good | 2 | \( 1 + 0.164T + 2T^{2} \) |
| 13 | \( 1 - 1.60iT - 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 5.56iT - 19T^{2} \) |
| 23 | \( 1 - 6.91iT - 23T^{2} \) |
| 29 | \( 1 + 9.80T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 + 0.866iT - 43T^{2} \) |
| 47 | \( 1 + 2.37iT - 47T^{2} \) |
| 53 | \( 1 - 6.59iT - 53T^{2} \) |
| 59 | \( 1 - 0.691iT - 59T^{2} \) |
| 61 | \( 1 + 9.81iT - 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 - 7.38iT - 71T^{2} \) |
| 73 | \( 1 - 16.3iT - 73T^{2} \) |
| 79 | \( 1 + 8.79iT - 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 + 7.69iT - 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−9.923521959196405321752447242215, −9.335974832469394242662724636239, −8.766846600578817397095344067307, −7.70885691191661401679041379464, −7.29296791924109028254346561837, −5.49002060722303857400114801345, −5.13502822492662225111282302032, −3.89373218105840596195260192240, −3.20722094144802507554176473998, −1.93933759906200316648131832033,
0.41116243822555043014335824925, 1.71862664586559881997051969834, 3.26484124387178617458085639508, 3.93113504153173367644963016770, 5.21312031779580658699907643133, 5.88939415628593162400979254149, 7.32594702715924736251142986004, 7.958519702148330708475358818186, 8.457043078547350815279043412140, 9.305198584577131226649350087589
