| L(s) = 1 | + 16·2-s + 256·4-s + 2.19e3i·5-s + 5.58e3·7-s + 4.09e3·8-s + 3.50e4i·10-s + 7.23e4i·11-s − 1.10e5i·13-s + 8.92e4·14-s + 6.55e4·16-s − 1.90e5i·17-s + (−4.81e5 + 3.01e5i)19-s + 5.60e5i·20-s + 1.15e6i·22-s + 1.44e6i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.56i·5-s + 0.878·7-s + 0.353·8-s + 1.10i·10-s + 1.48i·11-s − 1.06i·13-s + 0.621·14-s + 0.250·16-s − 0.553i·17-s + (−0.847 + 0.531i)19-s + 0.783i·20-s + 1.05i·22-s + 1.07i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.385i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.131537386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.131537386\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4.81e5 - 3.01e5i)T \) |
| good | 5 | \( 1 - 2.19e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 5.58e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.23e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + 1.10e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 1.90e5iT - 1.18e11T^{2} \) |
| 23 | \( 1 - 1.44e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 1.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.61e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.77e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 2.35e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.07e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.66e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 5.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.42e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.93e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.80e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.90e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.44e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.74e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 5.45e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 1.01e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.60e8iT - 7.60e17T^{2} \) |
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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
−10.44821448116826660151802059464, −9.904175162704160134875268403938, −8.184749063905144988377919171621, −7.33177769118148364491718521640, −6.69715866866248216135229274706, −5.51189715551142129125795606128, −4.55598207483025227499075996806, −3.39602316896939806368184652918, −2.49409052816342387871321880039, −1.54868454009867670626143444073,
0.41095985736454214221776003579, 1.34455900072572130326614308195, 2.39401253429309656704344483696, 4.06219505623889030650071845071, 4.54692623680456038039933205444, 5.57306718733746895253303441218, 6.39251322333159378081292120413, 7.923282881827699261899440505201, 8.574638808139910152043532402919, 9.314279930932798138599570971270
