| L(s) = 1 | + 1.41i·2-s + 2.34i·3-s − 2.00·4-s − 3.32·6-s + 7.61·7-s − 2.82i·8-s + 3.48·9-s + 12.3i·11-s − 4.69i·12-s − 13.0i·13-s + 10.7i·14-s + 4.00·16-s − 9.13·17-s + 4.92i·18-s − 14.4i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.782i·3-s − 0.500·4-s − 0.553·6-s + 1.08·7-s − 0.353i·8-s + 0.387·9-s + 1.12i·11-s − 0.391i·12-s − 1.00i·13-s + 0.769i·14-s + 0.250·16-s − 0.537·17-s + 0.273i·18-s − 0.760i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.192975720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192975720\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-4.51 - 22.5i)T \) |
| good | 3 | \( 1 - 2.34iT - 9T^{2} \) |
| 7 | \( 1 - 7.61T + 49T^{2} \) |
| 11 | \( 1 - 12.3iT - 121T^{2} \) |
| 13 | \( 1 + 13.0iT - 169T^{2} \) |
| 17 | \( 1 + 9.13T + 289T^{2} \) |
| 19 | \( 1 + 14.4iT - 361T^{2} \) |
| 29 | \( 1 - 21.2T + 841T^{2} \) |
| 31 | \( 1 - 36.8T + 961T^{2} \) |
| 37 | \( 1 - 56.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 70.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 70.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 82.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 23.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 69.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 25.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 28.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 69.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 45.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 74.4T + 9.40e3T^{2} \) |
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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
−9.755046023664239909381765899978, −9.168860468670768850953249659253, −8.076593550038675902122168969907, −7.56439467051787970041603749122, −6.63152302357706866909537983140, −5.46848212555828533494391673191, −4.64545340212120599265491679212, −4.29611782441143842692255206763, −2.77062809560411848706051861671, −1.22277909913099049910850646113,
0.78377605340829188770622017298, 1.71377397682663337275229837809, 2.67273251832991102438864531764, 4.10736539182979425487385938905, 4.74376411743225672901659953270, 6.03764238262390276404097855059, 6.77855869228829823369442094194, 7.956338365921815111499051025553, 8.365636400546476191513892297344, 9.282285639826969558281722460925
