| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (−0.977 + 0.212i)3-s + (−0.989 − 0.142i)4-s + (−0.931 − 2.03i)5-s + (−0.142 − 0.989i)6-s + (0.433 − 0.236i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (2.09 − 0.784i)10-s + (−3.22 − 2.79i)11-s + (0.997 − 0.0713i)12-s + (−2.44 + 4.47i)13-s + (0.205 + 0.449i)14-s + (1.34 + 1.78i)15-s + (0.959 + 0.281i)16-s + (5.62 + 4.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (−0.564 + 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.416 − 0.909i)5-s + (−0.0580 − 0.404i)6-s + (0.164 − 0.0895i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (0.662 − 0.247i)10-s + (−0.971 − 0.841i)11-s + (0.287 − 0.0205i)12-s + (−0.677 + 1.24i)13-s + (0.0548 + 0.120i)14-s + (0.346 + 0.461i)15-s + (0.239 + 0.0704i)16-s + (1.36 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.332965 + 0.622003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.332965 + 0.622003i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (0.931 + 2.03i)T \) |
| 23 | \( 1 + (-4.37 - 1.95i)T \) |
| good | 7 | \( 1 + (-0.433 + 0.236i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (3.22 + 2.79i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.44 - 4.47i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-5.62 - 4.21i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (1.01 - 7.03i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-4.61 + 0.662i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.0976 + 0.0627i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.92 + 1.46i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (3.92 - 8.58i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.62 + 7.47i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-0.867 - 0.867i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.53 - 8.31i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-1.73 - 5.92i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.51 + 3.90i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (1.43 + 0.102i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (0.855 + 0.987i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.66 - 4.89i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (6.31 - 1.85i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.64 + 15.1i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (7.84 + 5.04i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.44 - 17.2i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.60526756705176467282338427968, −9.907020038678550642278767678512, −8.839122390975383150256648877085, −8.089340432679688099707264093869, −7.41189883413184166699796644899, −6.13448369148283568342409809802, −5.42211593962057733749077874343, −4.59287977581035969077212501049, −3.55657194456075751969461851002, −1.34007705809226633650702836016,
0.44512283249380804116188088703, 2.49436614855325022183203502705, 3.18320598830579629770790929874, 4.84096435060750551746226837133, 5.27785335277440366161511822383, 6.87771933058071375010707142576, 7.44627414693808574588012305336, 8.372016135182034360244480963741, 9.785993006587941688473545435630, 10.28128029744363585578549135277
