| L(s) = 1 | + (−2.86 + 0.888i)3-s + 8.59i·5-s + 10.9·7-s + (7.41 − 5.09i)9-s − 2.75i·11-s + 4.43·13-s + (−7.63 − 24.6i)15-s + 25.4i·17-s + 17.5·19-s + (−31.3 + 9.71i)21-s − 17.5i·23-s − 48.8·25-s + (−16.7 + 21.1i)27-s + 19.6i·29-s + 2.58·31-s + ⋯ |
| L(s) = 1 | + (−0.955 + 0.296i)3-s + 1.71i·5-s + 1.56·7-s + (0.824 − 0.565i)9-s − 0.250i·11-s + 0.341·13-s + (−0.509 − 1.64i)15-s + 1.49i·17-s + 0.923·19-s + (−1.49 + 0.462i)21-s − 0.762i·23-s − 1.95·25-s + (−0.619 + 0.784i)27-s + 0.676i·29-s + 0.0833·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.837915 + 1.13722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.837915 + 1.13722i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.86 - 0.888i)T \) |
| good | 5 | \( 1 - 8.59iT - 25T^{2} \) |
| 7 | \( 1 - 10.9T + 49T^{2} \) |
| 11 | \( 1 + 2.75iT - 121T^{2} \) |
| 13 | \( 1 - 4.43T + 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 17.5T + 361T^{2} \) |
| 23 | \( 1 + 17.5iT - 529T^{2} \) |
| 29 | \( 1 - 19.6iT - 841T^{2} \) |
| 31 | \( 1 - 2.58T + 961T^{2} \) |
| 37 | \( 1 + 7.73T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 17.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 69.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 50.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 32.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 48.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 22.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 97.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 59.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 55.1T + 9.40e3T^{2} \) |
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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
−11.30941061420803876172843895294, −10.62555252121099706650023384685, −10.08156114592066056864320154206, −8.487667324699761436692715208981, −7.50217061326815434955189518449, −6.55560533881422489622986877148, −5.71804178690575212734046871403, −4.53165893161421804350463204059, −3.33224526725460347992639002618, −1.62265640277758325689736444328,
0.789527997211409017397839010510, 1.73903061592916153687498772563, 4.31666455742157984364052512234, 5.10615670484560280367563788567, 5.54596521943333442412683921070, 7.25249011763104016586366661942, 7.988285367882158667350106176896, 8.959735967617728245082026132279, 9.909912433114845537965755610581, 11.24568585107619765025431809405
