| L(s) = 1 | + (−0.453 − 2.22i)2-s + (−0.987 − 0.160i)3-s + (−2.88 + 1.22i)4-s + (1.01 − 0.250i)5-s + (0.0912 + 2.26i)6-s + (1.28 − 2.71i)7-s + (1.46 + 2.12i)8-s + (0.948 + 0.316i)9-s + (−1.01 − 2.14i)10-s + (0.829 − 0.276i)11-s + (3.04 − 0.750i)12-s + (−3.30 − 1.44i)13-s + (−6.61 − 1.62i)14-s + (−1.04 + 0.0841i)15-s + (−0.297 + 0.310i)16-s + (2.26 − 4.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.320 − 1.57i)2-s + (−0.569 − 0.0926i)3-s + (−1.44 + 0.614i)4-s + (0.454 − 0.111i)5-s + (0.0372 + 0.924i)6-s + (0.486 − 1.02i)7-s + (0.517 + 0.750i)8-s + (0.316 + 0.105i)9-s + (−0.321 − 0.677i)10-s + (0.250 − 0.0834i)11-s + (0.879 − 0.216i)12-s + (−0.916 − 0.400i)13-s + (−1.76 − 0.435i)14-s + (−0.269 + 0.0217i)15-s + (−0.0744 + 0.0775i)16-s + (0.548 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.272485 + 0.745729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272485 + 0.745729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 + 0.160i)T \) |
| 13 | \( 1 + (3.30 + 1.44i)T \) |
| good | 2 | \( 1 + (0.453 + 2.22i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-1.01 + 0.250i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 2.71i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-0.829 + 0.276i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-2.26 + 4.76i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (2.70 + 4.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.36 - 2.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.80 - 8.82i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (1.02 + 0.535i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-9.76 + 6.17i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (7.72 + 1.25i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (9.95 + 6.29i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-0.888 - 7.31i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (0.233 + 0.338i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (-1.95 - 2.04i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (2.35 + 0.190i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (3.75 + 1.60i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (-10.3 + 12.7i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-3.34 - 2.96i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (0.0267 + 0.220i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-4.51 + 11.8i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (3.50 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.41 + 4.88i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−10.52174297213973844981722558983, −9.777568131734146550414815724899, −9.081040735824418410534579774392, −7.71593995263861931180852840241, −6.82690801689933590098467507165, −5.27666158349550103535882635437, −4.43257912625519602915548245041, −3.16947332289686021388414832417, −1.82428524581352854310656242114, −0.56427771654068380215929799071,
2.09116280294580949488979104839, 4.30524650068000831468324075340, 5.31649739242656423200639388458, 6.09757931726276889752611852310, 6.59165028057378486693156386777, 8.032387095731568716142049676854, 8.332085314103502514050286984617, 9.675040900922986305964670135232, 10.08380692403316442315838775158, 11.62877888211307702022031247505
