| L(s) = 1 | + (0.621 + 1.27i)2-s + (−0.0272 − 1.73i)3-s + (−1.22 + 1.57i)4-s + (0.720 − 0.215i)5-s + (2.18 − 1.11i)6-s + (2.05 − 0.886i)7-s + (−2.76 − 0.575i)8-s + (−2.99 + 0.0944i)9-s + (0.721 + 0.780i)10-s + (4.86 − 1.15i)11-s + (2.76 + 2.08i)12-s + (3.66 + 2.41i)13-s + (2.40 + 2.05i)14-s + (−0.393 − 1.24i)15-s + (−0.990 − 3.87i)16-s + (−0.0735 − 0.202i)17-s + ⋯ |
| L(s) = 1 | + (0.439 + 0.898i)2-s + (−0.0157 − 0.999i)3-s + (−0.613 + 0.789i)4-s + (0.322 − 0.0964i)5-s + (0.891 − 0.453i)6-s + (0.777 − 0.335i)7-s + (−0.979 − 0.203i)8-s + (−0.999 + 0.0314i)9-s + (0.228 + 0.246i)10-s + (1.46 − 0.347i)11-s + (0.799 + 0.600i)12-s + (1.01 + 0.668i)13-s + (0.642 + 0.550i)14-s + (−0.101 − 0.320i)15-s + (−0.247 − 0.968i)16-s + (−0.0178 − 0.0490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73180 + 0.257099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73180 + 0.257099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.621 - 1.27i)T \) |
| 3 | \( 1 + (0.0272 + 1.73i)T \) |
| good | 5 | \( 1 + (-0.720 + 0.215i)T + (4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-2.05 + 0.886i)T + (4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-4.86 + 1.15i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-3.66 - 2.41i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (0.0735 + 0.202i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.348 + 0.958i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 4.92i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (6.14 - 0.358i)T + (28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-0.372 - 3.18i)T + (-30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (-2.60 - 2.18i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-0.104 - 0.208i)T + (-24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (4.41 - 4.16i)T + (2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (5.38 + 0.629i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (6.49 + 3.74i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (12.7 + 3.01i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-0.198 + 0.266i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (-2.69 - 0.157i)T + (66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (0.104 - 0.594i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.29 - 13.0i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-6.90 + 13.7i)T + (-47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (15.1 + 7.61i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (-13.6 + 2.41i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.45 + 11.5i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−11.68953129695989041926507056303, −11.22442748808302331322030733425, −9.294109955996901431349917529050, −8.619381879571861788764437183133, −7.70435523916482611075721880435, −6.63387808041442164276027742577, −6.11941835593487244819902191464, −4.78496893235118755444017187939, −3.51286178504399298418717052588, −1.49042802171419880571181408362,
1.71337549229496608849385698011, 3.38981138459059086746650158748, 4.23903578067956040641801686769, 5.40421182833320760423695497789, 6.18028296192191794494466229450, 8.158618189824153360794097084463, 9.211242250917410610288546415221, 9.714518508966347023691564155682, 10.89012579808446611763695969695, 11.38567710769736099509082673206
