L(s) = 1 | + (0.627 − 1.37i)2-s + (−0.959 − 0.281i)3-s + (−0.182 − 0.210i)4-s + (−0.841 + 0.540i)5-s + (−0.988 + 1.14i)6-s + (0.326 − 2.26i)7-s + (2.49 − 0.731i)8-s + (0.841 + 0.540i)9-s + (0.214 + 1.49i)10-s + (−1.70 − 3.73i)11-s + (0.115 + 0.253i)12-s + (−0.742 − 5.16i)13-s + (−2.90 − 1.86i)14-s + (0.959 − 0.281i)15-s + (0.637 − 4.43i)16-s + (−0.666 + 0.769i)17-s + ⋯ |
L(s) = 1 | + (0.443 − 0.971i)2-s + (−0.553 − 0.162i)3-s + (−0.0913 − 0.105i)4-s + (−0.376 + 0.241i)5-s + (−0.403 + 0.465i)6-s + (0.123 − 0.857i)7-s + (0.881 − 0.258i)8-s + (0.280 + 0.180i)9-s + (0.0679 + 0.472i)10-s + (−0.513 − 1.12i)11-s + (0.0334 + 0.0732i)12-s + (−0.206 − 1.43i)13-s + (−0.777 − 0.499i)14-s + (0.247 − 0.0727i)15-s + (0.159 − 1.10i)16-s + (−0.161 + 0.186i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.665284 - 1.21216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.665284 - 1.21216i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (1.25 + 4.62i)T \) |
good | 2 | \( 1 + (-0.627 + 1.37i)T + (-1.30 - 1.51i)T^{2} \) |
| 7 | \( 1 + (-0.326 + 2.26i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (1.70 + 3.73i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.742 + 5.16i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.666 - 0.769i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.23 - 4.88i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (2.57 - 2.97i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (1.25 - 0.369i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-9.07 - 5.83i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (2.13 - 1.37i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.93 - 2.03i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 + (1.10 - 7.67i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.655 + 4.55i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (7.98 - 2.34i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-5.63 + 12.3i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.47 + 3.23i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.560 - 0.646i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.86 - 12.9i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-8.08 - 5.19i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (5.51 + 1.61i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (8.11 - 5.21i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.01070588116361053539968575441, −10.75936755390509632264880647771, −9.909914961006044222869018036008, −8.007159509591817146424006155858, −7.61526713802362099176384064068, −6.20126990822355439152954682210, −5.02507776777809089832279143990, −3.78971121308655447641737277691, −2.88647046892851696866154723732, −0.952883982995784988382913987766,
2.08594790627458398427787664556, 4.28751644523045099255788391074, 5.03880321833658493675426248659, 5.88740007026910858704844247642, 7.06410164575362495443501045961, 7.59435460455338203809060156808, 9.056506870949397478484129157702, 9.817231596362790158674873946857, 11.24583394029430891145124848806, 11.69571133651647034501751960980