L(s) = 1 | + (−1.64 − 0.539i)3-s + (−1.30 + 1.81i)5-s + (−2.19 + 1.47i)7-s + (2.41 + 1.77i)9-s − 0.958i·11-s + 0.157·13-s + (3.12 − 2.29i)15-s + 2.51i·17-s − 1.98i·19-s + (4.41 − 1.23i)21-s − 2.67·23-s + (−1.60 − 4.73i)25-s + (−3.02 − 4.22i)27-s − 1.25i·29-s − 8.66i·31-s + ⋯ |
L(s) = 1 | + (−0.950 − 0.311i)3-s + (−0.582 + 0.812i)5-s + (−0.831 + 0.555i)7-s + (0.806 + 0.591i)9-s − 0.288i·11-s + 0.0436·13-s + (0.806 − 0.591i)15-s + 0.609i·17-s − 0.454i·19-s + (0.963 − 0.269i)21-s − 0.557·23-s + (−0.321 − 0.946i)25-s + (−0.582 − 0.813i)27-s − 0.232i·29-s − 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.185945 - 0.265526i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.185945 - 0.265526i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.64 + 0.539i)T \) |
| 5 | \( 1 + (1.30 - 1.81i)T \) |
| 7 | \( 1 + (2.19 - 1.47i)T \) |
good | 11 | \( 1 + 0.958iT - 11T^{2} \) |
| 13 | \( 1 - 0.157T + 13T^{2} \) |
| 17 | \( 1 - 2.51iT - 17T^{2} \) |
| 19 | \( 1 + 1.98iT - 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 1.25iT - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 2.29iT - 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 6.58iT - 43T^{2} \) |
| 47 | \( 1 - 5.60iT - 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.27iT - 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 - 9.75iT - 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 - 0.517T + 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 + 0.954T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.12864582685756942551923732070, −9.237272750139220353722953892632, −8.001758059149564321411298647188, −7.29097955451719295048116880925, −6.22219285419595182877553902082, −5.97164196720111378706901905181, −4.53542131103870693922083541535, −3.47868526499918746251576114957, −2.23366562150340092297221948852, −0.20127077126128006541075726348,
1.19652662058963379487213519963, 3.33339765203168010217672050605, 4.28716778768862804320201261759, 5.03942454671335980078937245879, 6.07610750964101946343148485239, 6.96476853593091384266300326005, 7.77716109420535470066162269849, 8.960308984153636824281024985675, 9.720009487984266202798268163571, 10.42965880686971597854318494496