L(s) = 1 | + (0.209 − 0.209i)3-s − 1.73i·7-s + 2.91i·9-s + (−0.505 − 0.505i)11-s + (1.88 − 1.88i)13-s − 4.53·17-s + (3.22 − 3.22i)19-s + (−0.364 − 0.364i)21-s − 8.85i·23-s + (1.23 + 1.23i)27-s + (−2.44 + 2.44i)29-s + 5.70·31-s − 0.211·33-s + (5.35 + 5.35i)37-s − 0.791i·39-s + ⋯ |
L(s) = 1 | + (0.120 − 0.120i)3-s − 0.656i·7-s + 0.970i·9-s + (−0.152 − 0.152i)11-s + (0.523 − 0.523i)13-s − 1.09·17-s + (0.738 − 0.738i)19-s + (−0.0794 − 0.0794i)21-s − 1.84i·23-s + (0.238 + 0.238i)27-s + (−0.453 + 0.453i)29-s + 1.02·31-s − 0.0368·33-s + (0.880 + 0.880i)37-s − 0.126i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.585506442\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585506442\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.209 + 0.209i)T - 3iT^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 + (0.505 + 0.505i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.88 + 1.88i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 19 | \( 1 + (-3.22 + 3.22i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.85iT - 23T^{2} \) |
| 29 | \( 1 + (2.44 - 2.44i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.0iT - 41T^{2} \) |
| 43 | \( 1 + (2.10 + 2.10i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 + (-1.37 - 1.37i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.64 + 6.64i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.26 + 5.26i)T - 61iT^{2} \) |
| 67 | \( 1 + (10.5 - 10.5i)T - 67iT^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 6.63iT - 73T^{2} \) |
| 79 | \( 1 + 4.27T + 79T^{2} \) |
| 83 | \( 1 + (-9.15 + 9.15i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.23iT - 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−9.076545061667461098599069293913, −8.466832309309860706460449024152, −7.66319624842674715005909720656, −6.93308038546577937537748357944, −6.08237218855534384878973303914, −4.96222011510499641232784822383, −4.35148633595140799082771656250, −3.10135748950688035322626498622, −2.16315193470187288006134555277, −0.65294235218499124810002173545,
1.31494039629517062340331993230, 2.59194015843742343510107465704, 3.63249621014968770658785474573, 4.43488209812067772511394785082, 5.66079164271156951819511036086, 6.19261978174283679952639244913, 7.16551481931912212269675767606, 8.029682963657187673678584013182, 8.959084341675952543259084730725, 9.424292175376866329529352397383