L(s) = 1 | − 2.64·3-s + 1.73i·7-s + 4.00·9-s + 3.46i·11-s − 4.58i·13-s + 3·17-s + (2.64 + 3.46i)19-s − 4.58i·21-s − 5.19i·23-s − 5·25-s − 2.64·27-s − 4.58i·29-s − 5.29·31-s − 9.16i·33-s + 9.16i·37-s + ⋯ |
L(s) = 1 | − 1.52·3-s + 0.654i·7-s + 1.33·9-s + 1.04i·11-s − 1.27i·13-s + 0.727·17-s + (0.606 + 0.794i)19-s − 0.999i·21-s − 1.08i·23-s − 25-s − 0.509·27-s − 0.850i·29-s − 0.950·31-s − 1.59i·33-s + 1.50i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5075708457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5075708457\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.64 - 3.46i)T \) |
good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 4.58iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 + 4.58iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 - 9.16iT - 37T^{2} \) |
| 41 | \( 1 - 9.16iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 4.58iT - 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 2.64T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 6.92iT - 83T^{2} \) |
| 89 | \( 1 - 18.3iT - 89T^{2} \) |
| 97 | \( 1 - 9.16iT - 97T^{2} \) |
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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
−10.11124384934224021788887968646, −9.563672415438273879934936898177, −8.178867372038408089827836804595, −7.53670387725448463751732062462, −6.43263947452194859177311187816, −5.75601524737962496426782283631, −5.18093220608518515839480448259, −4.22782658630816803377456899954, −2.80380046174846143033175828795, −1.29127988711736300540013461096,
0.29667533820549871869466008955, 1.57696037327253331610074079133, 3.45837092799349921579896549348, 4.33786618738200936847667330287, 5.50658236669242823059194275412, 5.80847557464670492280227643174, 7.05392149383546461651205617317, 7.34591530072929514340655078190, 8.786467651307475609210866677410, 9.550585049509338354911493652625