| L(s) = 1 | + (4.89 − 2.82i)5-s + (1.22 + 0.707i)11-s + 16·13-s + (14.6 + 8.48i)17-s + (4 + 6.92i)19-s + (11.0 − 6.36i)23-s + (3.49 − 6.06i)25-s + 24.0i·29-s + (−28 + 48.4i)31-s + (−12 − 20.7i)37-s − 50.9i·41-s + 40·43-s + (−9.79 + 5.65i)47-s + (37.9 + 21.9i)53-s + 8·55-s + ⋯ |
| L(s) = 1 | + (0.979 − 0.565i)5-s + (0.111 + 0.0642i)11-s + 1.23·13-s + (0.864 + 0.499i)17-s + (0.210 + 0.364i)19-s + (0.479 − 0.276i)23-s + (0.139 − 0.242i)25-s + 0.829i·29-s + (−0.903 + 1.56i)31-s + (−0.324 − 0.561i)37-s − 1.24i·41-s + 0.930·43-s + (−0.208 + 0.120i)47-s + (0.716 + 0.413i)53-s + 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0348i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.816520671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.816520671\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-4.89 + 2.82i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-1.22 - 0.707i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 16T + 169T^{2} \) |
| 17 | \( 1 + (-14.6 - 8.48i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-4 - 6.92i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-11.0 + 6.36i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 24.0iT - 841T^{2} \) |
| 31 | \( 1 + (28 - 48.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (12 + 20.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + (9.79 - 5.65i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-37.9 - 21.9i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-9.79 - 5.65i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20 - 34.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-13 + 22.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-44 + 76.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (41 + 71.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-53.8 + 31.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−8.872414537112956525655629165670, −8.717372493582598616057449194960, −7.51133310087648004439368678737, −6.66328953528414575042956567718, −5.63813802265193202195216738554, −5.38717600155067532924040329166, −4.06093063917143858565231820135, −3.20267383851181646111007704781, −1.80812511452159773967089065325, −1.08001271727007180904938594337,
0.906583973212363965835541656137, 2.08472106987046399145532655186, 3.06733446704945111946619073578, 3.99033770012522335989045990151, 5.23494538523713211645041889782, 5.96019452633570466330864877224, 6.55359782813601288012525299454, 7.51905705878129855087154738928, 8.308463077679554231223124088931, 9.358057531218304687748919197907
