L(s) = 1 | + (0.123 + 2.23i)5-s − 1.28i·7-s − 5.09·11-s − 3.56i·13-s − 0.895i·17-s + 5.34·19-s − 5.06i·23-s + (−4.96 + 0.551i)25-s − 3·29-s − 6.59·31-s + (2.87 − 0.159i)35-s − 7.24i·37-s + 7.84·41-s − 10.9i·43-s − 2.96i·47-s + ⋯ |
L(s) = 1 | + (0.0552 + 0.998i)5-s − 0.486i·7-s − 1.53·11-s − 0.990i·13-s − 0.217i·17-s + 1.22·19-s − 1.05i·23-s + (−0.993 + 0.110i)25-s − 0.557·29-s − 1.18·31-s + (0.486 − 0.0269i)35-s − 1.19i·37-s + 1.22·41-s − 1.66i·43-s − 0.433i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0552 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0552 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.018039421\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.018039421\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.123 - 2.23i)T \) |
good | 7 | \( 1 + 1.28iT - 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 3.56iT - 13T^{2} \) |
| 17 | \( 1 + 0.895iT - 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 5.06iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 7.24iT - 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 2.96iT - 47T^{2} \) |
| 53 | \( 1 - 4.78iT - 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 + 8.53iT - 67T^{2} \) |
| 71 | \( 1 - 5.34T + 71T^{2} \) |
| 73 | \( 1 - 9.34iT - 73T^{2} \) |
| 79 | \( 1 - 0.741T + 79T^{2} \) |
| 83 | \( 1 - 9.21iT - 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 3.45iT - 97T^{2} \) |
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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
−9.348486600867366428683051244718, −8.200465526629750689634099701397, −7.44479955199735619827075861915, −7.11268059782040398848743767231, −5.74674411489661892894476754456, −5.34290800776113372583499541035, −3.98588522556983369016279488951, −3.05575010292263130026729677582, −2.28867468010251078215337140843, −0.39336018644049372618428298458,
1.36768163352545050571213633341, 2.49354616142357128127107910198, 3.67903845425486228655264557835, 4.83545241774687972243416787526, 5.36654134870407329839726708256, 6.15243505484315799416401588218, 7.51369402946592631354108319526, 7.87647461999766586241613350238, 8.958319074165759453860853281352, 9.412242954173690778432367795422