| L(s) = 1 | − 1.65·2-s − 3.32·3-s + 0.743·4-s + 5.50·6-s − 2.81·7-s + 2.08·8-s + 8.06·9-s − 11-s − 2.47·12-s + 5.98·13-s + 4.65·14-s − 4.93·16-s + 6.49·17-s − 13.3·18-s − 19-s + 9.35·21-s + 1.65·22-s − 2.77·23-s − 6.92·24-s − 9.90·26-s − 16.8·27-s − 2.09·28-s − 1.78·29-s − 3.15·31-s + 4.01·32-s + 3.32·33-s − 10.7·34-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 1.92·3-s + 0.371·4-s + 2.24·6-s − 1.06·7-s + 0.735·8-s + 2.68·9-s − 0.301·11-s − 0.713·12-s + 1.65·13-s + 1.24·14-s − 1.23·16-s + 1.57·17-s − 3.14·18-s − 0.229·19-s + 2.04·21-s + 0.353·22-s − 0.577·23-s − 1.41·24-s − 1.94·26-s − 3.24·27-s − 0.395·28-s − 0.331·29-s − 0.567·31-s + 0.708·32-s + 0.579·33-s − 1.84·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 13 | \( 1 - 5.98T + 13T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 1.78T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 + 4.63T + 41T^{2} \) |
| 43 | \( 1 - 2.40T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 5.33T + 59T^{2} \) |
| 61 | \( 1 - 7.18T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 0.437T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 4.44T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 4.99T + 97T^{2} \) |
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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
−7.81611462424291358625211536353, −7.04400783224630035450988090521, −6.45846037344967725469358597337, −5.77587232729119860694577277506, −5.26250592461083963968837493915, −4.15189801884159778289470515593, −3.47796297777766996595124710141, −1.68360697737682742178596023455, −0.908605004971853356444654691527, 0,
0.908605004971853356444654691527, 1.68360697737682742178596023455, 3.47796297777766996595124710141, 4.15189801884159778289470515593, 5.26250592461083963968837493915, 5.77587232729119860694577277506, 6.45846037344967725469358597337, 7.04400783224630035450988090521, 7.81611462424291358625211536353
