| L(s) = 1 | + (1.26 − 0.460i)2-s + (2.70 − 0.984i)3-s + (−0.141 + 0.118i)4-s + (0.673 + 0.565i)5-s + (2.97 − 2.49i)6-s + (−1.47 + 2.54i)8-s + (4.05 − 3.40i)9-s + (1.11 + 0.405i)10-s + (1.11 + 1.92i)11-s + (−0.266 + 0.460i)12-s + (1.97 − 1.65i)13-s + (2.37 + 0.866i)15-s + (−0.624 + 3.54i)16-s + (0.358 + 0.300i)17-s + (3.56 − 6.17i)18-s + (2.77 + 3.35i)19-s + ⋯ |
| L(s) = 1 | + (0.895 − 0.325i)2-s + (1.56 − 0.568i)3-s + (−0.0707 + 0.0593i)4-s + (0.301 + 0.252i)5-s + (1.21 − 1.01i)6-s + (−0.520 + 0.901i)8-s + (1.35 − 1.13i)9-s + (0.352 + 0.128i)10-s + (0.335 + 0.581i)11-s + (−0.0768 + 0.133i)12-s + (0.546 − 0.458i)13-s + (0.614 + 0.223i)15-s + (−0.156 + 0.885i)16-s + (0.0869 + 0.0729i)17-s + (0.840 − 1.45i)18-s + (0.637 + 0.770i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.93842 - 0.839477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.93842 - 0.839477i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + (-2.77 - 3.35i)T \) |
| good | 2 | \( 1 + (-1.26 + 0.460i)T + (1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (-2.70 + 0.984i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.673 - 0.565i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 1.92i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.97 + 1.65i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.358 - 0.300i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.467 + 2.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.19 + 6.77i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 + (2.47 + 4.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.89 - 1.59i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.66 - 1.33i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (5.58 - 4.68i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (2.17 - 1.82i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (4.83 + 4.05i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.58 - 8.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.21 + 2.62i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.74 + 3.18i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.30 - 0.475i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (2.05 - 11.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.41 - 12.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.67 - 3.51i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.64 + 9.30i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.779391156353078276656737870814, −9.148699239495957869334490159050, −8.185723569348237222295746903631, −7.75764605374609261668934844981, −6.58204804248372760284674174096, −5.60294488214995667770131407181, −4.25763890619475400463400735457, −3.55105801616852685997559128149, −2.67259185438021378642036306117, −1.77324919451383477838692930606,
1.63462231734185639520291041377, 3.29379977364942042148502686329, 3.58958616896188215915044747859, 4.75903368363101350916921299218, 5.49693684544583918710580858928, 6.67622966747570022936039514527, 7.59213608371822573506327582661, 8.798549468412680609745565993987, 9.124192063072484505652969134331, 9.762573824255485458944224689612
