L(s) = 1 | + (−2 − 2i)3-s + (2 − 2i)7-s − i·9-s + 8·11-s + (−3 − 3i)13-s + (−7 + 7i)17-s − 20i·19-s − 8·21-s + (−2 − 2i)23-s + (−20 + 20i)27-s − 40i·29-s − 52·31-s + (−16 − 16i)33-s + (3 − 3i)37-s + 12i·39-s + ⋯ |
L(s) = 1 | + (−0.666 − 0.666i)3-s + (0.285 − 0.285i)7-s − 0.111i·9-s + 0.727·11-s + (−0.230 − 0.230i)13-s + (−0.411 + 0.411i)17-s − 1.05i·19-s − 0.380·21-s + (−0.0869 − 0.0869i)23-s + (−0.740 + 0.740i)27-s − 1.37i·29-s − 1.67·31-s + (−0.484 − 0.484i)33-s + (0.0810 − 0.0810i)37-s + 0.307i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.241484 - 0.850062i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.241484 - 0.850062i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2 + 2i)T + 9iT^{2} \) |
| 7 | \( 1 + (-2 + 2i)T - 49iT^{2} \) |
| 11 | \( 1 - 8T + 121T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 169iT^{2} \) |
| 17 | \( 1 + (7 - 7i)T - 289iT^{2} \) |
| 19 | \( 1 + 20iT - 361T^{2} \) |
| 23 | \( 1 + (2 + 2i)T + 529iT^{2} \) |
| 29 | \( 1 + 40iT - 841T^{2} \) |
| 31 | \( 1 + 52T + 961T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (42 + 42i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (18 - 18i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (53 + 53i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 20iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-62 + 62i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 28T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-47 - 47i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + (-18 - 18i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 80iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-63 + 63i)T - 9.40e3iT^{2} \) |
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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
−11.02925757659418387936956820616, −9.762342916850009284525604533544, −8.874035414996817492955701019736, −7.69332304342386810225971950094, −6.81494813700288390885837981447, −6.06944491546468071274347614546, −4.88041120016805143186330477747, −3.63743919069598103873364085530, −1.86793173718946404323648141080, −0.41548441986364967330235315004,
1.79004319827305602227911783764, 3.56679001993819847741093433042, 4.71301936461890810315053742308, 5.50709431715243810081971644691, 6.58802317176632074032570216965, 7.72371721650496784625870505845, 8.858393717482030118473741855376, 9.694841157114394839294440425383, 10.62568598103755953516354118666, 11.36219301202110009582046686494