L(s) = 1 | + (1.57 + 0.713i)3-s + (0.394 − 2.23i)5-s + (−3.88 + 3.25i)7-s + (1.98 + 2.25i)9-s + (0.823 + 4.67i)11-s + (4.81 − 1.75i)13-s + (2.21 − 3.25i)15-s + (−3.30 + 5.72i)17-s + (−1.30 − 2.26i)19-s + (−8.45 + 2.37i)21-s + (−0.771 − 0.647i)23-s + (−0.154 − 0.0561i)25-s + (1.52 + 4.96i)27-s + (4.01 + 1.46i)29-s + (2.07 + 1.73i)31-s + ⋯ |
L(s) = 1 | + (0.911 + 0.411i)3-s + (0.176 − 1.00i)5-s + (−1.46 + 1.23i)7-s + (0.660 + 0.750i)9-s + (0.248 + 1.40i)11-s + (1.33 − 0.486i)13-s + (0.573 − 0.839i)15-s + (−0.801 + 1.38i)17-s + (−0.299 − 0.518i)19-s + (−1.84 + 0.517i)21-s + (−0.160 − 0.135i)23-s + (−0.0308 − 0.0112i)25-s + (0.292 + 0.956i)27-s + (0.745 + 0.271i)29-s + (0.371 + 0.312i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53270 + 1.08059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53270 + 1.08059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.57 - 0.713i)T \) |
good | 5 | \( 1 + (-0.394 + 2.23i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (3.88 - 3.25i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.823 - 4.67i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.81 + 1.75i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.30 - 5.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.30 + 2.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.771 + 0.647i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.01 - 1.46i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.07 - 1.73i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.217 + 0.376i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.16 + 1.51i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.751 - 4.25i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.238 - 0.200i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + (-0.632 + 3.58i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.75 - 3.15i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-13.0 + 4.73i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.24 + 3.89i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.59 - 7.95i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.13 + 0.776i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (8.21 + 2.98i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (4.16 + 7.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.71 + 9.74i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.990987780328018017917580837025, −9.360127676869527643100659376315, −8.733300390023275384030932943813, −8.257242062477069034192728499007, −6.76914688452215518712345423653, −6.03072983462974562543059803229, −4.85932243279223305480112045818, −3.95297958046480167742086748146, −2.85972871758211775558995340911, −1.74507586356072259434503954612,
0.847313654015675730014952868279, 2.69976747288528785549558891603, 3.41082024120191076982052303347, 4.08918384477230654178850481102, 6.29302595149113876188451817076, 6.46934289625917007019426166398, 7.28732444807013135019381716400, 8.334187182508986922869224119602, 9.202994283006748093079102326629, 9.910450934585033949492510328624