| L(s) = 1 | + 0.381·2-s − 1.85·4-s − 1.47·8-s + 3.14·16-s + 8.23·17-s + 4.70·19-s + 7.47·23-s − 10.7·31-s + 4.14·32-s + 3.14·34-s + 1.79·38-s + 2.85·46-s + 8.94·47-s − 7·49-s + 9.76·53-s + 12.4·61-s − 4.09·62-s − 4.70·64-s − 15.2·68-s − 8.72·76-s − 1.29·79-s − 5.94·83-s − 13.8·92-s + 3.41·94-s − 2.67·98-s + 3.72·106-s − 17.8·107-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.927·4-s − 0.520·8-s + 0.786·16-s + 1.99·17-s + 1.08·19-s + 1.55·23-s − 1.92·31-s + 0.732·32-s + 0.539·34-s + 0.291·38-s + 0.420·46-s + 1.30·47-s − 49-s + 1.34·53-s + 1.58·61-s − 0.519·62-s − 0.588·64-s − 1.85·68-s − 1.00·76-s − 0.145·79-s − 0.652·83-s − 1.44·92-s + 0.352·94-s − 0.270·98-s + 0.362·106-s − 1.72·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.431704364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431704364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 8.23T + 17T^{2} \) |
| 19 | \( 1 - 4.70T + 19T^{2} \) |
| 23 | \( 1 - 7.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 9.76T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 1.29T + 79T^{2} \) |
| 83 | \( 1 + 5.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.35512206999629643908977223765, −9.567299117901375406968753704114, −8.914917417765390268653735458889, −7.87665575801581715244869725539, −7.09471470718208076698673689061, −5.55479613634857707357429754257, −5.25941550531456978974929114273, −3.87692393850453507716821001926, −3.07858435171798324011693354819, −1.05639703435404280964920717849,
1.05639703435404280964920717849, 3.07858435171798324011693354819, 3.87692393850453507716821001926, 5.25941550531456978974929114273, 5.55479613634857707357429754257, 7.09471470718208076698673689061, 7.87665575801581715244869725539, 8.914917417765390268653735458889, 9.567299117901375406968753704114, 10.35512206999629643908977223765
