| L(s) = 1 | + (1.28 + 0.600i)2-s + (−1.09 − 1.32i)3-s + (1.27 + 1.53i)4-s + (0.153 − 2.23i)5-s + (−0.608 − 2.35i)6-s + (3.86 − 1.25i)7-s + (0.714 + 2.73i)8-s + (0.00760 − 0.0398i)9-s + (1.53 − 2.76i)10-s + (−0.775 + 6.13i)11-s + (0.635 − 3.38i)12-s + (1.21 − 6.39i)13-s + (5.70 + 0.713i)14-s + (−3.12 + 2.24i)15-s + (−0.728 + 3.93i)16-s + (0.702 + 2.73i)17-s + ⋯ |
| L(s) = 1 | + (0.905 + 0.424i)2-s + (−0.633 − 0.765i)3-s + (0.639 + 0.768i)4-s + (0.0685 − 0.997i)5-s + (−0.248 − 0.961i)6-s + (1.46 − 0.474i)7-s + (0.252 + 0.967i)8-s + (0.00253 − 0.0132i)9-s + (0.485 − 0.874i)10-s + (−0.233 + 1.84i)11-s + (0.183 − 0.976i)12-s + (0.338 − 1.77i)13-s + (1.52 + 0.190i)14-s + (−0.806 + 0.579i)15-s + (−0.182 + 0.983i)16-s + (0.170 + 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48138 - 0.947872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48138 - 0.947872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.600i)T \) |
| 5 | \( 1 + (-0.153 + 2.23i)T \) |
| good | 3 | \( 1 + (1.09 + 1.32i)T + (-0.562 + 2.94i)T^{2} \) |
| 7 | \( 1 + (-3.86 + 1.25i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (0.775 - 6.13i)T + (-10.6 - 2.73i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 6.39i)T + (-12.0 - 4.78i)T^{2} \) |
| 17 | \( 1 + (-0.702 - 2.73i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (1.94 + 1.61i)T + (3.56 + 18.6i)T^{2} \) |
| 23 | \( 1 + (-4.00 + 4.26i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (2.00 - 0.126i)T + (28.7 - 3.63i)T^{2} \) |
| 31 | \( 1 + (-7.65 + 1.96i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (1.30 + 0.718i)T + (19.8 + 31.2i)T^{2} \) |
| 41 | \( 1 + (5.57 - 5.23i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-5.92 + 4.30i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (2.12 + 5.37i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (-1.58 + 2.49i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (-9.90 + 4.66i)T + (37.6 - 45.4i)T^{2} \) |
| 61 | \( 1 + (6.90 - 7.34i)T + (-3.83 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-0.0945 + 1.50i)T + (-66.4 - 8.39i)T^{2} \) |
| 71 | \( 1 + (3.26 - 1.29i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-3.60 - 1.69i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (1.11 + 1.34i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (1.57 - 1.90i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (3.96 - 8.41i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-7.77 + 0.488i)T + (96.2 - 12.1i)T^{2} \) |
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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
−10.13559953049057519334880831802, −8.591039346839361448898795073147, −7.941525441692733857161719719521, −7.34126996927290763227110926766, −6.40461473029091217246367131103, −5.32340625727244326686357264176, −4.88503377965576937984026620944, −4.01498240532831888874820224400, −2.21344011918493690098111726722, −1.11399198826120750335657131537,
1.65038021110382908283285379113, 2.85818686591401727808339353387, 3.95229032287906761148360252033, 4.81461893920096755638789790486, 5.60897850587530259067710697907, 6.26797211075412921832634218828, 7.34833332312642376470288077663, 8.479733084571967074227551468721, 9.506977642768253174990516403190, 10.55398111908223027960858329475
