L(s) = 1 | + (−0.477 − 1.33i)2-s + (−1.84 + 1.06i)3-s + (−1.54 + 1.27i)4-s + (0.5 − 0.866i)5-s + (2.29 + 1.94i)6-s + (−2.17 − 1.50i)7-s + (2.42 + 1.45i)8-s + (0.759 − 1.31i)9-s + (−1.39 − 0.252i)10-s + (2.04 + 3.54i)11-s + (1.49 − 3.98i)12-s + 4.95·13-s + (−0.966 + 3.61i)14-s + 2.12i·15-s + (0.772 − 3.92i)16-s + (2.09 − 1.20i)17-s + ⋯ |
L(s) = 1 | + (−0.337 − 0.941i)2-s + (−1.06 + 0.613i)3-s + (−0.772 + 0.635i)4-s + (0.223 − 0.387i)5-s + (0.936 + 0.793i)6-s + (−0.822 − 0.569i)7-s + (0.858 + 0.512i)8-s + (0.253 − 0.438i)9-s + (−0.440 − 0.0798i)10-s + (0.616 + 1.06i)11-s + (0.431 − 1.14i)12-s + 1.37·13-s + (−0.258 + 0.966i)14-s + 0.548i·15-s + (0.193 − 0.981i)16-s + (0.507 − 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.709456 - 0.0796085i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.709456 - 0.0796085i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.477 + 1.33i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.17 + 1.50i)T \) |
good | 3 | \( 1 + (1.84 - 1.06i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 + (-2.09 + 1.20i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.22 - 3.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.443 + 0.255i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.08iT - 29T^{2} \) |
| 31 | \( 1 + (-2.30 - 3.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.64 - 4.99i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.81iT - 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + (-0.698 + 1.21i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.79 - 3.92i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.82 + 5.09i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.33 - 2.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 6.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.54iT - 71T^{2} \) |
| 73 | \( 1 + (-6.99 + 4.03i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.21 - 1.85i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.94iT - 83T^{2} \) |
| 89 | \( 1 + (3.81 + 2.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.67iT - 97T^{2} \) |
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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
−11.74540396080241436869127088434, −10.83969228344080980459704835705, −9.946135541834408182007157043291, −9.560032499596916891552087225587, −8.225297174531135959606525847789, −6.81894314853238614437627266945, −5.54350020416876127628196508332, −4.44352920439935326937831508725, −3.42924773582807374344805610428, −1.19996821332825791721482324192,
0.912633733133135056946500130768, 3.51419904741960915676540591067, 5.42878771657357577966944952698, 6.22428605516544267404156600569, 6.49968454088718616770292602432, 7.86643219100450739128888794456, 8.981672622446999925302777144740, 9.827626921939527314803105203798, 11.10771144983368655254890084531, 11.70227100080053506384595055523