L(s) = 1 | + (233. − 233. i)2-s + (−4.70e3 − 4.70e3i)3-s − 4.31e4i·4-s + (−3.87e5 + 4.54e4i)5-s − 2.19e6·6-s + (−2.16e5 + 2.16e5i)7-s + (5.20e6 + 5.20e6i)8-s + 1.28e6i·9-s + (−7.98e7 + 1.01e8i)10-s − 2.53e8·11-s + (−2.03e8 + 2.03e8i)12-s + (−7.01e8 − 7.01e8i)13-s + 1.00e8i·14-s + (2.04e9 + 1.61e9i)15-s + 5.26e9·16-s + (1.16e9 − 1.16e9i)17-s + ⋯ |
L(s) = 1 | + (0.910 − 0.910i)2-s + (−0.717 − 0.717i)3-s − 0.659i·4-s + (−0.993 + 0.116i)5-s − 1.30·6-s + (−0.0375 + 0.0375i)7-s + (0.310 + 0.310i)8-s + 0.0298i·9-s + (−0.798 + 1.01i)10-s − 1.18·11-s + (−0.472 + 0.472i)12-s + (−0.860 − 0.860i)13-s + 0.0683i·14-s + (0.796 + 0.629i)15-s + 1.22·16-s + (0.166 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(0.209376 + 0.942989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.209376 + 0.942989i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (3.87e5 - 4.54e4i)T \) |
good | 2 | \( 1 + (-233. + 233. i)T - 6.55e4iT^{2} \) |
| 3 | \( 1 + (4.70e3 + 4.70e3i)T + 4.30e7iT^{2} \) |
| 7 | \( 1 + (2.16e5 - 2.16e5i)T - 3.32e13iT^{2} \) |
| 11 | \( 1 + 2.53e8T + 4.59e16T^{2} \) |
| 13 | \( 1 + (7.01e8 + 7.01e8i)T + 6.65e17iT^{2} \) |
| 17 | \( 1 + (-1.16e9 + 1.16e9i)T - 4.86e19iT^{2} \) |
| 19 | \( 1 + 3.15e10iT - 2.88e20T^{2} \) |
| 23 | \( 1 + (-2.72e10 - 2.72e10i)T + 6.13e21iT^{2} \) |
| 29 | \( 1 + 2.93e11iT - 2.50e23T^{2} \) |
| 31 | \( 1 + 2.25e11T + 7.27e23T^{2} \) |
| 37 | \( 1 + (4.39e12 - 4.39e12i)T - 1.23e25iT^{2} \) |
| 41 | \( 1 - 1.29e13T + 6.37e25T^{2} \) |
| 43 | \( 1 + (1.86e12 + 1.86e12i)T + 1.36e26iT^{2} \) |
| 47 | \( 1 + (-1.81e13 + 1.81e13i)T - 5.66e26iT^{2} \) |
| 53 | \( 1 + (1.95e13 + 1.95e13i)T + 3.87e27iT^{2} \) |
| 59 | \( 1 + 1.51e14iT - 2.15e28T^{2} \) |
| 61 | \( 1 + 6.79e13T + 3.67e28T^{2} \) |
| 67 | \( 1 + (-6.76e13 + 6.76e13i)T - 1.64e29iT^{2} \) |
| 71 | \( 1 + 2.56e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + (7.68e14 + 7.68e14i)T + 6.50e29iT^{2} \) |
| 79 | \( 1 + 1.48e15iT - 2.30e30T^{2} \) |
| 83 | \( 1 + (-1.97e15 - 1.97e15i)T + 5.07e30iT^{2} \) |
| 89 | \( 1 + 2.36e15iT - 1.54e31T^{2} \) |
| 97 | \( 1 + (3.97e15 - 3.97e15i)T - 6.14e31iT^{2} \) |
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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
−19.19675996209597287528881563746, −17.53083233150028423018793710384, −15.33639211178947480195518493741, −13.14356162477693823710301616812, −12.12450702252496307383710284477, −10.93650895372630602805423842284, −7.49670541338625454604816538948, −5.02198372615212692129361508528, −2.88954408001752653467619848818, −0.39641366595429618069398260963,
4.17469482125694636681492872773, 5.43411015184544664196208611284, 7.54858609782699029410654367019, 10.52070156875966359097470458471, 12.44106769493788917224181862541, 14.50966421653476819478244695435, 15.91215657933600922697864974132, 16.60750670246183011707753426761, 19.09998697542673814602583410005, 21.22187302783207582381876570026