| L(s) = 1 | + 3.65i·2-s + 2.64·4-s − 45.6i·5-s + 34.3i·7-s + 68.1i·8-s + 166.·10-s + 103.·11-s + 134.·13-s − 125.·14-s − 206.·16-s − 240.·17-s − 100. i·19-s − 120. i·20-s + 376. i·22-s + 475.·23-s + ⋯ |
| L(s) = 1 | + 0.913i·2-s + 0.165·4-s − 1.82i·5-s + 0.700i·7-s + 1.06i·8-s + 1.66·10-s + 0.852·11-s + 0.798·13-s − 0.640·14-s − 0.807·16-s − 0.831·17-s − 0.279i·19-s − 0.301i·20-s + 0.778i·22-s + 0.899·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.548470576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.548470576\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-1.74e3 + 619. i)T \) |
| good | 2 | \( 1 - 3.65iT - 16T^{2} \) |
| 5 | \( 1 + 45.6iT - 625T^{2} \) |
| 7 | \( 1 - 34.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 103.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 134.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 240.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 100. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 475.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 159. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.11e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.45e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.06e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 2.93e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 825.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.24e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 6.20e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 265.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.86e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.27e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 4.95e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.35e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + 5.35e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.19e3T + 8.85e7T^{2} \) |
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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
−10.92381808416010955976499991185, −9.185055054978605588375223423895, −8.883337101569147778600211211425, −8.123308847588744737783468232262, −6.88582782324175525964446731989, −5.88674765547343029352154640386, −5.16087275945442999576015012945, −4.11751719653150797732275948095, −2.17195038929045578339873809234, −0.880696327777674589899255986533,
1.08321772904586050219929221390, 2.43658549041062159510528519134, 3.35541572530407998343220571552, 4.15798399462381972156340552618, 6.34189781064133631924143097206, 6.68368764491758570838213886972, 7.62692794957708710705950764760, 9.130269145580753204571214006973, 10.29854517795831491030958994716, 10.61344943678733582059824811600
