| L(s) = 1 | + (2.23 − 2.23i)3-s + (−2.23 − 2.23i)7-s − 7.00i·9-s − 10.0·21-s + (6.70 − 6.70i)23-s + (−8.94 − 8.94i)27-s + 6i·29-s − 12·41-s + (−2.23 + 2.23i)43-s + (−6.70 − 6.70i)47-s + 3.00i·49-s − 8·61-s + (−15.6 + 15.6i)63-s + (11.1 + 11.1i)67-s − 30.0i·69-s + ⋯ |
| L(s) = 1 | + (1.29 − 1.29i)3-s + (−0.845 − 0.845i)7-s − 2.33i·9-s − 2.18·21-s + (1.39 − 1.39i)23-s + (−1.72 − 1.72i)27-s + 1.11i·29-s − 1.87·41-s + (−0.340 + 0.340i)43-s + (−0.978 − 0.978i)47-s + 0.428i·49-s − 1.02·61-s + (−1.97 + 1.97i)63-s + (1.36 + 1.36i)67-s − 3.61i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138833643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138833643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-2.23 + 2.23i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.23 + 2.23i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (-6.70 + 6.70i)T - 23iT^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + (2.23 - 2.23i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.70 + 6.70i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (-11.1 - 11.1i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-6.70 + 6.70i)T - 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−8.745317933887126052803531283058, −8.434827604038177592443756923405, −7.28034061006716998686749649107, −6.94075640828951657934291303632, −6.29935147146270531966716659586, −4.81285716207612743894987948430, −3.49926226400355369727595460109, −3.04305013922393692705711668468, −1.85059286920267894722971885122, −0.68065253205299105847164602824,
2.04530559482849895487284918821, 3.11583879702628824115326895184, 3.49094016899079719483234096769, 4.68558965207510125871950860003, 5.39534190248615249142837333260, 6.49658393546939259146040478205, 7.63304384116467567464772599213, 8.379010613957239173646645083055, 9.185788614370790979489714186462, 9.496105630132233981715491433904
