L(s) = 1 | + (−0.929 − 0.368i)2-s + (0.740 + 0.170i)3-s + (0.728 + 0.684i)4-s + (−0.625 − 0.430i)6-s + (−0.425 − 0.904i)8-s + (−0.380 − 0.184i)9-s + (0.0562 − 1.49i)11-s + (0.422 + 0.630i)12-s + (0.0627 + 0.998i)16-s + (−1.28 − 0.983i)17-s + (0.285 + 0.312i)18-s + (−1.15 − 1.55i)19-s + (−0.601 + 1.36i)22-s + (−0.161 − 0.742i)24-s + (0.968 − 0.248i)25-s + ⋯ |
L(s) = 1 | + (−0.929 − 0.368i)2-s + (0.740 + 0.170i)3-s + (0.728 + 0.684i)4-s + (−0.625 − 0.430i)6-s + (−0.425 − 0.904i)8-s + (−0.380 − 0.184i)9-s + (0.0562 − 1.49i)11-s + (0.422 + 0.630i)12-s + (0.0627 + 0.998i)16-s + (−1.28 − 0.983i)17-s + (0.285 + 0.312i)18-s + (−1.15 − 1.55i)19-s + (−0.601 + 1.36i)22-s + (−0.161 − 0.742i)24-s + (0.968 − 0.248i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7425493267\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7425493267\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.929 + 0.368i)T \) |
| 251 | \( 1 + (-0.356 - 0.934i)T \) |
good | 3 | \( 1 + (-0.740 - 0.170i)T + (0.899 + 0.437i)T^{2} \) |
| 5 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 7 | \( 1 + (0.745 + 0.666i)T^{2} \) |
| 11 | \( 1 + (-0.0562 + 1.49i)T + (-0.997 - 0.0753i)T^{2} \) |
| 13 | \( 1 + (-0.938 + 0.344i)T^{2} \) |
| 17 | \( 1 + (1.28 + 0.983i)T + (0.260 + 0.965i)T^{2} \) |
| 19 | \( 1 + (1.15 + 1.55i)T + (-0.285 + 0.958i)T^{2} \) |
| 23 | \( 1 + (0.778 + 0.627i)T^{2} \) |
| 29 | \( 1 + (-0.920 + 0.391i)T^{2} \) |
| 31 | \( 1 + (0.236 + 0.971i)T^{2} \) |
| 37 | \( 1 + (0.0376 + 0.999i)T^{2} \) |
| 41 | \( 1 + (0.389 - 0.347i)T + (0.112 - 0.993i)T^{2} \) |
| 43 | \( 1 + (-0.472 - 1.94i)T + (-0.888 + 0.459i)T^{2} \) |
| 47 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 53 | \( 1 + (-0.162 + 0.986i)T^{2} \) |
| 59 | \( 1 + (-0.981 + 0.751i)T + (0.260 - 0.965i)T^{2} \) |
| 61 | \( 1 + (0.711 - 0.702i)T^{2} \) |
| 67 | \( 1 + (-1.90 + 0.191i)T + (0.979 - 0.199i)T^{2} \) |
| 71 | \( 1 + (0.556 - 0.830i)T^{2} \) |
| 73 | \( 1 + (0.0941 - 1.06i)T + (-0.984 - 0.175i)T^{2} \) |
| 79 | \( 1 + (0.514 + 0.857i)T^{2} \) |
| 83 | \( 1 + (-1.29 - 0.197i)T + (0.954 + 0.297i)T^{2} \) |
| 89 | \( 1 + (-0.518 - 1.74i)T + (-0.837 + 0.546i)T^{2} \) |
| 97 | \( 1 + (-1.10 - 1.09i)T + (0.0125 + 0.999i)T^{2} \) |
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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
−9.063234961637799988382351929608, −8.515765171908830145429964543343, −8.015766952651559646467063110516, −6.71741902630202249883177780011, −6.43122585832464270737831943569, −4.98164171802062742433688845366, −3.84710043329822454520037913471, −2.87822457276146687709912357046, −2.41021363658907532972782969340, −0.61537904801381748214048186960,
1.87284797366472767075865806272, 2.23020265464374478276083951264, 3.68848605654352539345628981537, 4.76912993943209786315654100209, 5.85216606786071807647076003149, 6.67330435603463311969833434946, 7.36537320749453878954557559817, 8.166815410938172926875082615908, 8.706750455980361469798935744772, 9.285473167491038819500496168447