L(s) = 1 | + (−205. + 205. i)2-s + (−8.38e3 − 8.38e3i)3-s − 1.85e4i·4-s + (−6.50e4 − 3.85e5i)5-s + 3.44e6·6-s + (−4.20e6 + 4.20e6i)7-s + (−9.62e6 − 9.62e6i)8-s + 9.77e7i·9-s + (9.23e7 + 6.56e7i)10-s + 1.64e8·11-s + (−1.55e8 + 1.55e8i)12-s + (2.05e8 + 2.05e8i)13-s − 1.72e9i·14-s + (−2.68e9 + 3.77e9i)15-s + 5.16e9·16-s + (2.00e9 − 2.00e9i)17-s + ⋯ |
L(s) = 1 | + (−0.801 + 0.801i)2-s + (−1.27 − 1.27i)3-s − 0.283i·4-s + (−0.166 − 0.986i)5-s + 2.04·6-s + (−0.728 + 0.728i)7-s + (−0.573 − 0.573i)8-s + 2.27i·9-s + (0.923 + 0.656i)10-s + 0.765·11-s + (−0.362 + 0.362i)12-s + (0.251 + 0.251i)13-s − 1.16i·14-s + (−1.04 + 1.47i)15-s + 1.20·16-s + (0.287 − 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(0.332202 + 0.223518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332202 + 0.223518i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (6.50e4 + 3.85e5i)T \) |
good | 2 | \( 1 + (205. - 205. i)T - 6.55e4iT^{2} \) |
| 3 | \( 1 + (8.38e3 + 8.38e3i)T + 4.30e7iT^{2} \) |
| 7 | \( 1 + (4.20e6 - 4.20e6i)T - 3.32e13iT^{2} \) |
| 11 | \( 1 - 1.64e8T + 4.59e16T^{2} \) |
| 13 | \( 1 + (-2.05e8 - 2.05e8i)T + 6.65e17iT^{2} \) |
| 17 | \( 1 + (-2.00e9 + 2.00e9i)T - 4.86e19iT^{2} \) |
| 19 | \( 1 + 6.44e9iT - 2.88e20T^{2} \) |
| 23 | \( 1 + (-2.07e10 - 2.07e10i)T + 6.13e21iT^{2} \) |
| 29 | \( 1 - 7.06e11iT - 2.50e23T^{2} \) |
| 31 | \( 1 + 3.56e11T + 7.27e23T^{2} \) |
| 37 | \( 1 + (-2.11e11 + 2.11e11i)T - 1.23e25iT^{2} \) |
| 41 | \( 1 - 8.10e12T + 6.37e25T^{2} \) |
| 43 | \( 1 + (6.40e12 + 6.40e12i)T + 1.36e26iT^{2} \) |
| 47 | \( 1 + (2.97e12 - 2.97e12i)T - 5.66e26iT^{2} \) |
| 53 | \( 1 + (-8.09e13 - 8.09e13i)T + 3.87e27iT^{2} \) |
| 59 | \( 1 - 1.64e13iT - 2.15e28T^{2} \) |
| 61 | \( 1 - 1.82e14T + 3.67e28T^{2} \) |
| 67 | \( 1 + (4.64e14 - 4.64e14i)T - 1.64e29iT^{2} \) |
| 71 | \( 1 + 9.33e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + (-8.72e14 - 8.72e14i)T + 6.50e29iT^{2} \) |
| 79 | \( 1 + 5.24e14iT - 2.30e30T^{2} \) |
| 83 | \( 1 + (-1.59e15 - 1.59e15i)T + 5.07e30iT^{2} \) |
| 89 | \( 1 + 5.35e15iT - 1.54e31T^{2} \) |
| 97 | \( 1 + (6.01e15 - 6.01e15i)T - 6.14e31iT^{2} \) |
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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
−19.32166379312465613981493318225, −18.14621149810066406636369131253, −16.95353989655844778142450532563, −16.10459335528811840731195319439, −12.87811192065241421099429062486, −11.91522046664174144927890698724, −8.937609186991686365952893728785, −7.13528559000091076648955769744, −5.78417264509509327307395924178, −0.996232177040116314764800138517,
0.41931919893275409725806967857, 3.66519258132799312901559417841, 6.18369301974784976561432232958, 9.737361915736190198564016379712, 10.57799134037337712394631246827, 11.66879319787202060165138365464, 14.93676856051539899374808162225, 16.61281428389725496005597227231, 17.86098035679897946402536602099, 19.46011054161238149142265189172