| L(s) = 1 | + (−0.157 + 0.895i)2-s + (0.719 − 1.24i)3-s + (1.10 + 0.401i)4-s + (−0.683 − 3.87i)5-s + (1.00 + 0.841i)6-s + (−2.04 + 3.54i)7-s + (−1.44 + 2.49i)8-s + (0.463 + 0.803i)9-s + 3.58·10-s + (0.168 + 0.954i)11-s + (1.29 − 1.08i)12-s + (−3.23 − 2.71i)13-s + (−2.85 − 2.39i)14-s + (−5.32 − 1.93i)15-s + (−0.213 − 0.179i)16-s + (0.420 + 0.727i)17-s + ⋯ |
| L(s) = 1 | + (−0.111 + 0.633i)2-s + (0.415 − 0.719i)3-s + (0.551 + 0.200i)4-s + (−0.305 − 1.73i)5-s + (0.409 + 0.343i)6-s + (−0.773 + 1.33i)7-s + (−0.510 + 0.883i)8-s + (0.154 + 0.267i)9-s + 1.13·10-s + (0.0507 + 0.287i)11-s + (0.373 − 0.313i)12-s + (−0.897 − 0.753i)13-s + (−0.761 − 0.639i)14-s + (−1.37 − 0.500i)15-s + (−0.0533 − 0.0448i)16-s + (0.101 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.999913 + 0.0537810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999913 + 0.0537810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (-4.68 + 7.14i)T \) |
| good | 2 | \( 1 + (0.157 - 0.895i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.719 + 1.24i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.683 + 3.87i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.04 - 3.54i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.168 - 0.954i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (3.23 + 2.71i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.420 - 0.727i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.367 - 2.08i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (0.820 + 4.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.27 + 7.25i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.87 + 2.14i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.599 - 3.39i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (6.12 - 5.13i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.526 + 0.911i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.700 - 0.587i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.193 + 1.09i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (0.904 + 0.758i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-7.16 + 6.01i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.736 + 0.617i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.338 - 1.91i)T + (-66.7 - 24.2i)T^{2} \) |
| 79 | \( 1 + (-9.51 - 7.98i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 + (2.73 + 2.29i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.87 + 4.97i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.98914156735687728209520193605, −13.27838801490401379199062477488, −12.35050061236537396318296768961, −12.06956781356592117090442305534, −9.741118233335446470270308026866, −8.388860822718222438947646113312, −7.974719935314392229470643340443, −6.33978349499128497560187421705, −5.05599998005466270208110926207, −2.39808806077895397509096866568,
2.97862477306642642444763176417, 3.82806513713963177383688377061, 6.72928841063970649160283582592, 7.13264432923833181990868858425, 9.566281098432179408497465801531, 10.25868874472550490626800725157, 10.93923034795930826975572022549, 12.07072025547620358127659914291, 13.78134843924277496526036786910, 14.64801528692617994489663919430
