| L(s) = 1 | + 1.21·2-s + 2.90·3-s − 0.525·4-s − 5-s + 3.52·6-s + 1.52·7-s − 3.06·8-s + 5.42·9-s − 1.21·10-s − 4.90·11-s − 1.52·12-s − 6.42·13-s + 1.85·14-s − 2.90·15-s − 2.67·16-s − 2.14·17-s + 6.59·18-s − 2.28·19-s + 0.525·20-s + 4.42·21-s − 5.95·22-s + 6.90·23-s − 8.90·24-s + 25-s − 7.80·26-s + 7.05·27-s − 0.801·28-s + ⋯ |
| L(s) = 1 | + 0.858·2-s + 1.67·3-s − 0.262·4-s − 0.447·5-s + 1.43·6-s + 0.576·7-s − 1.08·8-s + 1.80·9-s − 0.384·10-s − 1.47·11-s − 0.440·12-s − 1.78·13-s + 0.495·14-s − 0.749·15-s − 0.668·16-s − 0.520·17-s + 1.55·18-s − 0.523·19-s + 0.117·20-s + 0.966·21-s − 1.26·22-s + 1.43·23-s − 1.81·24-s + 0.200·25-s − 1.53·26-s + 1.35·27-s − 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 6.42T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 31 | \( 1 + 1.71T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.37T + 53T^{2} \) |
| 59 | \( 1 + 3.18T + 59T^{2} \) |
| 61 | \( 1 - 2.42T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 - 3.57T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 0.341T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−8.101926118374928217031855706401, −7.44494593997879591809928467340, −6.83820624517000479484291550358, −5.37090459241885913174232389590, −4.82164686105692823699127435320, −4.31522064007672728190090361480, −3.21697391197670304149041584058, −2.78918519951561115477274122299, −1.98685110061312099981051352536, 0,
1.98685110061312099981051352536, 2.78918519951561115477274122299, 3.21697391197670304149041584058, 4.31522064007672728190090361480, 4.82164686105692823699127435320, 5.37090459241885913174232389590, 6.83820624517000479484291550358, 7.44494593997879591809928467340, 8.101926118374928217031855706401
