L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.292 − 0.507i)5-s + (−0.499 − 0.866i)9-s + (−2.41 + 4.18i)11-s + 4.24·13-s + 0.585·15-s + (−2.29 + 3.97i)17-s + (0.585 + 1.01i)19-s + (0.414 + 0.717i)23-s + (2.32 − 4.03i)25-s + 0.999·27-s − 2.82·29-s + (1.41 − 2.44i)31-s + (−2.41 − 4.18i)33-s + (−4.82 − 8.36i)37-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.130 − 0.226i)5-s + (−0.166 − 0.288i)9-s + (−0.727 + 1.26i)11-s + 1.17·13-s + 0.151·15-s + (−0.556 + 0.963i)17-s + (0.134 + 0.232i)19-s + (0.0863 + 0.149i)23-s + (0.465 − 0.806i)25-s + 0.192·27-s − 0.525·29-s + (0.254 − 0.439i)31-s + (−0.420 − 0.727i)33-s + (−0.793 − 1.37i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7619790944\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7619790944\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.292 + 0.507i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + (2.29 - 3.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.585 - 1.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.414 - 0.717i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + (-1.41 + 2.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + (-6.24 - 10.8i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.24 - 7.34i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 2.65i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.65 - 9.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + (8.12 - 14.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.17 - 2.02i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (7.12 + 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−9.245291421853021582240124758469, −8.634608875112943652444719296713, −7.83942015845333795140660487831, −6.98719465704319850376056273991, −6.08462351024484500938163650188, −5.38838712614704163833718543131, −4.39357527582527998993836045578, −3.90795473054530167592804226747, −2.63005227129464675761450515195, −1.45752086628693038239251068919,
0.27393437285472496430538448632, 1.53170232092274488988036953495, 2.90494128923396826989141726653, 3.49929254197255194792370720493, 4.87671875123341616288257802917, 5.52377100147487369750127577983, 6.42857366738149936235763965929, 6.98805732770594990253239100287, 7.943308881939195830906801671996, 8.576466728698745796521616782203