L(s) = 1 | − 2.37i·3-s − 0.792i·5-s + 3.31·7-s − 2.62·9-s + 3.37i·11-s − 4.10i·13-s − 1.87·15-s + 5·17-s − i·19-s − 7.86i·21-s + 2.52·23-s + 4.37·25-s − 0.883i·27-s − 5.98i·29-s − 3.46·31-s + ⋯ |
L(s) = 1 | − 1.36i·3-s − 0.354i·5-s + 1.25·7-s − 0.875·9-s + 1.01i·11-s − 1.13i·13-s − 0.485·15-s + 1.21·17-s − 0.229i·19-s − 1.71i·21-s + 0.526·23-s + 0.874·25-s − 0.169i·27-s − 1.11i·29-s − 0.622·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.953614297\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.953614297\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + iT \) |
good | 3 | \( 1 + 2.37iT - 3T^{2} \) |
| 5 | \( 1 + 0.792iT - 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 3.37iT - 11T^{2} \) |
| 13 | \( 1 + 4.10iT - 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 5.98iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 3.16iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.37iT - 43T^{2} \) |
| 47 | \( 1 + 9.30T + 47T^{2} \) |
| 53 | \( 1 - 9.45iT - 53T^{2} \) |
| 59 | \( 1 - 4.37iT - 59T^{2} \) |
| 61 | \( 1 + 4.55iT - 61T^{2} \) |
| 67 | \( 1 + 8.37iT - 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 - 8.74T + 97T^{2} \) |
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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
−9.404979390538569676937055665047, −8.311172809260085018376733797215, −7.75130529194113109539068316317, −7.33395295329179552912041371936, −6.21080104298761349758029663455, −5.27592433553822880920106061045, −4.52649546749383743508688380858, −2.94361977816303228611969455357, −1.78878215865209644962971452947, −0.962322197851378193396681577292,
1.52143263540541849284876611229, 3.13392879742463735289517653005, 3.86819229434019105338563217548, 4.93254056768507418073777074947, 5.36460901845087189259938218961, 6.62109541026131474306171422011, 7.62522358784906160477305884846, 8.640604967150968629236848381858, 9.058625037482359487585022235353, 10.14314906167616906118186308215