L(s) = 1 | + 1.53·2-s + 0.360·4-s − 2.16·5-s + 2.96·7-s − 2.51·8-s − 3.32·10-s + 11-s + 1.11·13-s + 4.55·14-s − 4.59·16-s − 6.92·17-s + 6.93·19-s − 0.780·20-s + 1.53·22-s + 0.602·23-s − 0.303·25-s + 1.71·26-s + 1.06·28-s − 4.99·29-s − 1.25·31-s − 2.01·32-s − 10.6·34-s − 6.42·35-s − 3.94·37-s + 10.6·38-s + 5.45·40-s + 1.36·41-s + ⋯ |
L(s) = 1 | + 1.08·2-s + 0.180·4-s − 0.969·5-s + 1.12·7-s − 0.890·8-s − 1.05·10-s + 0.301·11-s + 0.310·13-s + 1.21·14-s − 1.14·16-s − 1.67·17-s + 1.59·19-s − 0.174·20-s + 0.327·22-s + 0.125·23-s − 0.0606·25-s + 0.336·26-s + 0.201·28-s − 0.926·29-s − 0.224·31-s − 0.356·32-s − 1.82·34-s − 1.08·35-s − 0.648·37-s + 1.72·38-s + 0.863·40-s + 0.213·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.93T + 19T^{2} \) |
| 23 | \( 1 - 0.602T + 23T^{2} \) |
| 29 | \( 1 + 4.99T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 - 1.36T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 + 4.08T + 59T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 + 6.01T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 3.58T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.57334588046699844737973274727, −7.07922293460276345850298316174, −6.05984358058107506027207406730, −5.41484138019693614197961444886, −4.62758583600298502452296631542, −4.19753755458656201582033643276, −3.51946475956294723072889219644, −2.60038319175842598232822593336, −1.43974673249606540165431078647, 0,
1.43974673249606540165431078647, 2.60038319175842598232822593336, 3.51946475956294723072889219644, 4.19753755458656201582033643276, 4.62758583600298502452296631542, 5.41484138019693614197961444886, 6.05984358058107506027207406730, 7.07922293460276345850298316174, 7.57334588046699844737973274727