| L(s) = 1 | + 2.26·2-s + 2.20·3-s + 3.14·4-s − 5-s + 4.99·6-s − 7-s + 2.58·8-s + 1.84·9-s − 2.26·10-s − 1.87·11-s + 6.91·12-s − 6.18·13-s − 2.26·14-s − 2.20·15-s − 0.414·16-s − 6.31·17-s + 4.19·18-s + 3.74·19-s − 3.14·20-s − 2.20·21-s − 4.25·22-s + 1.97·23-s + 5.69·24-s + 25-s − 14.0·26-s − 2.53·27-s − 3.14·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.27·3-s + 1.57·4-s − 0.447·5-s + 2.03·6-s − 0.377·7-s + 0.914·8-s + 0.615·9-s − 0.717·10-s − 0.565·11-s + 1.99·12-s − 1.71·13-s − 0.606·14-s − 0.568·15-s − 0.103·16-s − 1.53·17-s + 0.987·18-s + 0.859·19-s − 0.702·20-s − 0.480·21-s − 0.907·22-s + 0.410·23-s + 1.16·24-s + 0.200·25-s − 2.75·26-s − 0.488·27-s − 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 3 | \( 1 - 2.20T + 3T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 + 6.31T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 - 1.97T + 23T^{2} \) |
| 29 | \( 1 - 0.131T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 + 0.581T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 - 2.41T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 + 3.42T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 + 8.39T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 2.99T + 97T^{2} \) |
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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
−7.39518911267628133625388520654, −6.89260319085531524672488884726, −5.91896042631292411109913165381, −5.17803931680698564907870019792, −4.49093589821953145828086782187, −3.96171847179389094117357949315, −2.98528306613488084214295253935, −2.73242014157813359350173836052, −2.00177302243030477604240012834, 0,
2.00177302243030477604240012834, 2.73242014157813359350173836052, 2.98528306613488084214295253935, 3.96171847179389094117357949315, 4.49093589821953145828086782187, 5.17803931680698564907870019792, 5.91896042631292411109913165381, 6.89260319085531524672488884726, 7.39518911267628133625388520654
