L(s) = 1 | + (1.17 + 0.780i)2-s + (−1.51 − 0.848i)3-s + (0.780 + 1.84i)4-s + (−1.11 − 2.17i)6-s + 3.02i·7-s + (−0.516 + 2.78i)8-s + (1.56 + 2.56i)9-s − 1.32·11-s + (0.382 − 3.44i)12-s + 5.12·13-s + (−2.35 + 3.56i)14-s + (−2.78 + 2.87i)16-s + 2i·17-s + (−0.158 + 4.23i)18-s + 1.32i·19-s + ⋯ |
L(s) = 1 | + (0.833 + 0.552i)2-s + (−0.871 − 0.489i)3-s + (0.390 + 0.920i)4-s + (−0.456 − 0.889i)6-s + 1.14i·7-s + (−0.182 + 0.983i)8-s + (0.520 + 0.853i)9-s − 0.399·11-s + (0.110 − 0.993i)12-s + 1.42·13-s + (−0.630 + 0.951i)14-s + (−0.695 + 0.718i)16-s + 0.485i·17-s + (−0.0374 + 0.999i)18-s + 0.303i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15417 + 1.03307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15417 + 1.03307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.17 - 0.780i)T \) |
| 3 | \( 1 + (1.51 + 0.848i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.02iT - 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 1.32iT - 19T^{2} \) |
| 23 | \( 1 - 0.371T + 23T^{2} \) |
| 29 | \( 1 + 3.12iT - 29T^{2} \) |
| 31 | \( 1 + 4.71iT - 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.73iT - 43T^{2} \) |
| 47 | \( 1 - 3.02T + 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 4.34iT - 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 8.10iT - 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−11.98493974508743832886436001244, −11.45665198057408352252726551329, −10.41687627316335975887985570989, −8.742424477736436642054486288316, −7.983371522542226013442311084443, −6.74551305057028987246536200450, −5.88454904558691989480645404344, −5.31220684544641650870305082249, −3.85823842921416065547660384256, −2.17596323524311128721225686552,
1.07838627622742878055959292716, 3.35621918157467965496976573047, 4.29584253173765608915211451174, 5.28431288016760838679428698690, 6.35538263924068519824575907287, 7.24028958102532479004651994047, 8.999079707261716256282494779966, 10.20287699958123065425384378148, 10.74039560232277892652826018185, 11.37322350519567213477611741103