L(s) = 1 | + (0.947 + 1.04i)2-s + (−0.204 + 1.98i)4-s + i·5-s + (2.29 + 1.31i)7-s + (−2.28 + 1.66i)8-s + (−1.04 + 0.947i)10-s + 0.477i·11-s + 2.96i·13-s + (0.796 + 3.65i)14-s + (−3.91 − 0.814i)16-s − 3.83i·17-s + 5.31·19-s + (−1.98 − 0.204i)20-s + (−0.500 + 0.452i)22-s + 7.60i·23-s + ⋯ |
L(s) = 1 | + (0.669 + 0.742i)2-s + (−0.102 + 0.994i)4-s + 0.447i·5-s + (0.868 + 0.496i)7-s + (−0.807 + 0.590i)8-s + (−0.332 + 0.299i)10-s + 0.143i·11-s + 0.821i·13-s + (0.212 + 0.977i)14-s + (−0.979 − 0.203i)16-s − 0.929i·17-s + 1.21·19-s + (−0.444 − 0.0457i)20-s + (−0.106 + 0.0963i)22-s + 1.58i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.341449765\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.341449765\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.947 - 1.04i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.29 - 1.31i)T \) |
good | 11 | \( 1 - 0.477iT - 11T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 + 3.83iT - 17T^{2} \) |
| 19 | \( 1 - 5.31T + 19T^{2} \) |
| 23 | \( 1 - 7.60iT - 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 1.01iT - 41T^{2} \) |
| 43 | \( 1 + 6.85iT - 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 9.30T + 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 + 4.75iT - 61T^{2} \) |
| 67 | \( 1 - 1.30iT - 67T^{2} \) |
| 71 | \( 1 - 9.18iT - 71T^{2} \) |
| 73 | \( 1 - 4.49iT - 73T^{2} \) |
| 79 | \( 1 - 8.80iT - 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 1.60iT - 97T^{2} \) |
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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
−9.766661208767502129841301423742, −9.095447595304270868394156419885, −8.193582089055735260451289330890, −7.31369728212946396481995432423, −6.90988657619961191678851261780, −5.55288344482460288030940668957, −5.23045104704533005359191940677, −4.07035409933140851797483255638, −3.11579291028749059712155878224, −1.91068930049639494765294222666,
0.813144125612419738781755113393, 1.91513878378910681799511781183, 3.21924333972862801957377753083, 4.15091959760602348938001453171, 5.01058141788821154008877652117, 5.66559088782973175927866223689, 6.73537271327176269848038439994, 7.86643585474287998554021200176, 8.575926101345314569721162871097, 9.554909870900612311103929192628