| L(s) = 1 | + (0.625 + 1.92i)2-s + (−0.809 + 0.587i)3-s + (−1.70 + 1.23i)4-s + (0.958 + 2.02i)5-s + (−1.63 − 1.19i)6-s + 7-s + (−0.166 − 0.121i)8-s + (0.309 − 0.951i)9-s + (−3.29 + 3.11i)10-s + (0.187 + 0.578i)11-s + (0.649 − 1.99i)12-s + (−1.19 + 3.67i)13-s + (0.625 + 1.92i)14-s + (−1.96 − 1.07i)15-s + (−1.17 + 3.60i)16-s + (−5.86 − 4.26i)17-s + ⋯ |
| L(s) = 1 | + (0.442 + 1.36i)2-s + (−0.467 + 0.339i)3-s + (−0.850 + 0.617i)4-s + (0.428 + 0.903i)5-s + (−0.668 − 0.485i)6-s + 0.377·7-s + (−0.0589 − 0.0427i)8-s + (0.103 − 0.317i)9-s + (−1.04 + 0.983i)10-s + (0.0566 + 0.174i)11-s + (0.187 − 0.577i)12-s + (−0.330 + 1.01i)13-s + (0.167 + 0.514i)14-s + (−0.506 − 0.276i)15-s + (−0.292 + 0.900i)16-s + (−1.42 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00503000 - 1.64903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00503000 - 1.64903i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.958 - 2.02i)T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + (-0.625 - 1.92i)T + (-1.61 + 1.17i)T^{2} \) |
| 11 | \( 1 + (-0.187 - 0.578i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.19 - 3.67i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (5.86 + 4.26i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.96 - 4.33i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.55 + 4.78i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-8.37 + 6.08i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.25 + 1.64i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.00 + 3.08i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.107 - 0.329i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 9.26T + 43T^{2} \) |
| 47 | \( 1 + (-3.04 + 2.21i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.22 + 3.07i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.77 - 5.47i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.17 - 9.76i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-12.1 - 8.83i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (4.83 - 3.51i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.498 - 1.53i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 8.19i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.88 - 6.45i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.243 - 0.748i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.02 + 3.64i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.37956129487173998037197205051, −10.34789555260699774448748341758, −9.523720968059433795205972425348, −8.403770347333925816163231757522, −7.24178754664358167192630907088, −6.73784843748670512759510150931, −5.90123654876145356761549204300, −4.93056728869761826875339241358, −4.08824803409358802407722978112, −2.29833311666677313197537688246,
0.940421915797849676923132743840, 2.04072712059969548859159756340, 3.37430772137729238373191179764, 4.78932280087315929331411547903, 5.22473204559544947563939716088, 6.58648407422095638081650940241, 7.84409126083452438784257246525, 8.879419948475685533552926342380, 9.835108068766820055628233384177, 10.67006093521026567068469308578
