L(s) = 1 | + (−0.866 − 0.5i)3-s + (−0.866 − 2.5i)7-s + (−1 − 1.73i)9-s + (−3 + 5.19i)11-s + 2i·13-s + (−5.19 − 3i)17-s + (4 + 6.92i)19-s + (−0.500 + 2.59i)21-s + (2.59 − 1.5i)23-s + 5i·27-s − 3·29-s + (−1 + 1.73i)31-s + (5.19 − 3i)33-s + (−6.92 + 4i)37-s + (1 − 1.73i)39-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.288i)3-s + (−0.327 − 0.944i)7-s + (−0.333 − 0.577i)9-s + (−0.904 + 1.56i)11-s + 0.554i·13-s + (−1.26 − 0.727i)17-s + (0.917 + 1.58i)19-s + (−0.109 + 0.566i)21-s + (0.541 − 0.312i)23-s + 0.962i·27-s − 0.557·29-s + (−0.179 + 0.311i)31-s + (0.904 − 0.522i)33-s + (−1.13 + 0.657i)37-s + (0.160 − 0.277i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0984958 + 0.221436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0984958 + 0.221436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 3 | \( 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.75198239684200011976679816588, −9.908808765299322568805362941954, −9.276519288571159531942200011853, −7.966259883329521821766315631821, −7.06370523361702915577230407559, −6.61210937356520402815524906658, −5.30768511409901177601284251375, −4.43450097858714459320649525872, −3.22838470323681510562797052722, −1.67106804890505790399825074717,
0.12857275858498555271702710163, 2.43051744339291963771087778339, 3.33116525699630870877050130879, 4.98274831252056889652300491816, 5.51708881811588923150174182914, 6.32452547653195436148859367660, 7.59029294036418735482971328542, 8.615253523776414758279115496443, 9.075307495591748469278006426491, 10.36370115533847598153936931895