| L(s) = 1 | + 2-s − 2.81·3-s + 4-s − 5-s − 2.81·6-s − 2.15·7-s + 8-s + 4.93·9-s − 10-s − 6.40·11-s − 2.81·12-s − 1.28·13-s − 2.15·14-s + 2.81·15-s + 16-s − 2.94·17-s + 4.93·18-s − 3.06·19-s − 20-s + 6.06·21-s − 6.40·22-s − 6.87·23-s − 2.81·24-s + 25-s − 1.28·26-s − 5.45·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.62·3-s + 0.5·4-s − 0.447·5-s − 1.15·6-s − 0.814·7-s + 0.353·8-s + 1.64·9-s − 0.316·10-s − 1.93·11-s − 0.813·12-s − 0.357·13-s − 0.575·14-s + 0.727·15-s + 0.250·16-s − 0.714·17-s + 1.16·18-s − 0.704·19-s − 0.223·20-s + 1.32·21-s − 1.36·22-s − 1.43·23-s − 0.575·24-s + 0.200·25-s − 0.252·26-s − 1.05·27-s − 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08362364186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08362364186\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 + 6.40T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 + 3.06T + 19T^{2} \) |
| 23 | \( 1 + 6.87T + 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 + 9.82T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.167T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 + 4.65T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 9.02T + 83T^{2} \) |
| 89 | \( 1 - 6.22T + 89T^{2} \) |
| 97 | \( 1 - 9.19T + 97T^{2} \) |
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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
−7.58010600511909856318644502361, −7.34477412301647845632685234089, −6.36584220975135301127728538218, −5.84302291137961367359863246757, −5.34375264130189566100231570840, −4.55735448012236690179197583590, −3.94764781766158664528867567814, −2.85445293967137785650020136595, −1.93983884431261210164934033172, −0.13718741074643768028761156934,
0.13718741074643768028761156934, 1.93983884431261210164934033172, 2.85445293967137785650020136595, 3.94764781766158664528867567814, 4.55735448012236690179197583590, 5.34375264130189566100231570840, 5.84302291137961367359863246757, 6.36584220975135301127728538218, 7.34477412301647845632685234089, 7.58010600511909856318644502361
