| L(s) = 1 | + (0.365 − 0.930i)2-s + (0.0747 − 0.997i)3-s + (−0.733 − 0.680i)4-s + (2.56 + 1.75i)5-s + (−0.900 − 0.433i)6-s + (1.92 + 1.81i)7-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (2.56 − 1.75i)10-s + (4.48 − 0.675i)11-s + (−0.733 + 0.680i)12-s + (−0.915 − 1.14i)13-s + (2.39 − 1.12i)14-s + (1.93 − 2.43i)15-s + (0.0747 + 0.997i)16-s + (−6.10 − 1.88i)17-s + ⋯ |
| L(s) = 1 | + (0.258 − 0.658i)2-s + (0.0431 − 0.575i)3-s + (−0.366 − 0.340i)4-s + (1.14 + 0.783i)5-s + (−0.367 − 0.177i)6-s + (0.727 + 0.686i)7-s + (−0.318 + 0.153i)8-s + (−0.329 − 0.0496i)9-s + (0.812 − 0.553i)10-s + (1.35 − 0.203i)11-s + (−0.211 + 0.196i)12-s + (−0.254 − 0.318i)13-s + (0.639 − 0.301i)14-s + (0.500 − 0.627i)15-s + (0.0186 + 0.249i)16-s + (−1.48 − 0.456i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58293 - 0.800092i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58293 - 0.800092i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.365 + 0.930i)T \) |
| 3 | \( 1 + (-0.0747 + 0.997i)T \) |
| 7 | \( 1 + (-1.92 - 1.81i)T \) |
| good | 5 | \( 1 + (-2.56 - 1.75i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-4.48 + 0.675i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (0.915 + 1.14i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (6.10 + 1.88i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (0.00666 + 0.0115i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (8.06 - 2.48i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (1.14 + 5.00i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.347 + 0.601i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.30 + 4.91i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-2.35 + 1.13i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.56 - 3.64i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (4.22 - 10.7i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (8.22 + 7.63i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (7.34 - 5.00i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-0.0791 + 0.0734i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (2.92 - 5.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.95 + 12.9i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-3.36 - 8.56i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (4.25 + 7.37i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.17 - 3.98i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (6.56 + 0.989i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 0.717T + 97T^{2} \) |
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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
−11.53411767305126570337208016960, −11.01783775389627616221100796240, −9.687627020608109764515775950613, −9.108534231879561721279645906887, −7.80170831882802719588382186639, −6.35799635369731446935666455971, −5.86892528203679176822713969225, −4.32545965261398348004340153341, −2.61604336155491754762157977240, −1.79483099901049583614989498599,
1.80192704338350413816435269359, 4.11228489664563472664145155769, 4.69342853656397109971479239012, 5.93725979969433966109191668300, 6.81003913015241665425590534588, 8.254072998134726770940800943128, 9.071374736053452826071053121779, 9.784592969609858472608059342933, 10.90007067031484832132815173480, 12.01195520758817411635845376825
