| L(s) = 1 | + (0.618 + 1.61i)3-s + i·5-s + 4.20i·7-s + (−2.23 + 2.00i)9-s − 11-s + 1.36·13-s + (−1.61 + 0.618i)15-s − 2.20i·17-s + 1.62i·19-s + (−6.80 + 2.60i)21-s − 5.20·23-s − 25-s + (−4.61 − 2.38i)27-s + 0.371i·29-s + 7.20i·31-s + ⋯ |
| L(s) = 1 | + (0.356 + 0.934i)3-s + 0.447i·5-s + 1.59i·7-s + (−0.745 + 0.666i)9-s − 0.301·11-s + 0.378·13-s + (−0.417 + 0.159i)15-s − 0.535i·17-s + 0.373i·19-s + (−1.48 + 0.567i)21-s − 1.08·23-s − 0.200·25-s + (−0.888 − 0.458i)27-s + 0.0689i·29-s + 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.297558402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297558402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4.20iT - 7T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 + 2.20iT - 17T^{2} \) |
| 19 | \( 1 - 1.62iT - 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 - 0.371iT - 29T^{2} \) |
| 31 | \( 1 - 7.20iT - 31T^{2} \) |
| 37 | \( 1 + 0.0213T + 37T^{2} \) |
| 41 | \( 1 + 3.62iT - 41T^{2} \) |
| 43 | \( 1 - 5.47iT - 43T^{2} \) |
| 47 | \( 1 - 3.96T + 47T^{2} \) |
| 53 | \( 1 + 0.0345iT - 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 + 0.798T + 61T^{2} \) |
| 67 | \( 1 + 8.41iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 - 1.95iT - 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−9.399749696832010123949895557494, −8.454098414017025227629159899812, −8.176110793372239986039668532070, −6.98063256772178150234811498635, −5.92945768685158629474068981846, −5.47741249186811278595295212878, −4.59453335869804399976208175469, −3.53249291214319116741784186037, −2.79206195586334672533755020431, −2.00252856434794041419028358406,
0.39821122029245242620122152421, 1.35651307412744657691515809676, 2.42769896921142298007178416703, 3.71019333077412949615137409747, 4.20059428634516804902437175527, 5.44100717620530208951935486923, 6.31109266491811245901861150589, 6.99607150973478455939218304075, 7.80857998645194361632358071123, 8.135886911258754920520668684767
