L(s) = 1 | + (1.73 + 0.0417i)3-s + 5-s + 0.963i·7-s + (2.99 + 0.144i)9-s − 6.53·11-s − 4.46i·13-s + (1.73 + 0.0417i)15-s − 6.10i·17-s + 2.66·19-s + (−0.0402 + 1.66i)21-s + 7.66i·23-s + 25-s + (5.18 + 0.375i)27-s + 3.54i·29-s − 9.54i·31-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0241i)3-s + 0.447·5-s + 0.364i·7-s + (0.998 + 0.0481i)9-s − 1.97·11-s − 1.23i·13-s + (0.447 + 0.0107i)15-s − 1.48i·17-s + 0.611·19-s + (−0.00877 + 0.363i)21-s + 1.59i·23-s + 0.200·25-s + (0.997 + 0.0722i)27-s + 0.658i·29-s − 1.71i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.732391536\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.732391536\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.73 - 0.0417i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (6.02 + 5.53i)T \) |
good | 7 | \( 1 - 0.963iT - 7T^{2} \) |
| 11 | \( 1 + 6.53T + 11T^{2} \) |
| 13 | \( 1 + 4.46iT - 13T^{2} \) |
| 17 | \( 1 + 6.10iT - 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 - 7.66iT - 23T^{2} \) |
| 29 | \( 1 - 3.54iT - 29T^{2} \) |
| 31 | \( 1 + 9.54iT - 31T^{2} \) |
| 37 | \( 1 - 9.79T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 6.22iT - 43T^{2} \) |
| 47 | \( 1 + 9.79iT - 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 9.79iT - 59T^{2} \) |
| 61 | \( 1 + 4.55iT - 61T^{2} \) |
| 71 | \( 1 - 3.73iT - 71T^{2} \) |
| 73 | \( 1 - 0.167T + 73T^{2} \) |
| 79 | \( 1 + 6.48iT - 79T^{2} \) |
| 83 | \( 1 + 17.0iT - 83T^{2} \) |
| 89 | \( 1 - 6.38iT - 89T^{2} \) |
| 97 | \( 1 + 5.76iT - 97T^{2} \) |
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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
−8.184640587951438402168745486828, −7.60332753713622893327699257396, −7.38805225024187434390060553050, −5.92503964210660904457432174899, −5.36410681520938769951234370369, −4.71534698626937737576344738633, −3.39169107094882764333457061150, −2.77094759231605330492289369424, −2.22091127859772316857777393605, −0.71467714442490960890396086118,
1.19466086922389228712499951346, 2.37283837167322686064351096550, 2.75478680720715275452692345716, 4.04658414842461862624354826359, 4.53226658582849472891827296420, 5.58208817283408638027260997455, 6.41815882164785079836089322092, 7.24671688803102239231515027489, 7.86259985333916320955311047551, 8.525200393969676192804162539350