| L(s) = 1 | + 1.93·2-s + 1.74·4-s − 4.18·5-s − 7-s − 0.491·8-s − 8.10·10-s + 11-s − 3.17·13-s − 1.93·14-s − 4.44·16-s − 6.85·17-s − 0.318·19-s − 7.31·20-s + 1.93·22-s + 1.87·23-s + 12.5·25-s − 6.14·26-s − 1.74·28-s + 3.17·29-s + 9.23·31-s − 7.61·32-s − 13.2·34-s + 4.18·35-s − 7.55·37-s − 0.616·38-s + 2.06·40-s − 9.36·41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.872·4-s − 1.87·5-s − 0.377·7-s − 0.173·8-s − 2.56·10-s + 0.301·11-s − 0.880·13-s − 0.517·14-s − 1.11·16-s − 1.66·17-s − 0.0731·19-s − 1.63·20-s + 0.412·22-s + 0.390·23-s + 2.51·25-s − 1.20·26-s − 0.329·28-s + 0.589·29-s + 1.65·31-s − 1.34·32-s − 2.27·34-s + 0.708·35-s − 1.24·37-s − 0.100·38-s + 0.325·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 5 | \( 1 + 4.18T + 5T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 6.85T + 17T^{2} \) |
| 19 | \( 1 + 0.318T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 9.23T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 + 9.36T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 + 0.508T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 - 5.01T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.31892881686011013076334032188, −8.936466461282421859104530205200, −8.266687228494963366525791727658, −6.95467925429763141050855852171, −6.65750338714462598945729693110, −5.02634003713131708178759643517, −4.45121417313434353743955773395, −3.63280981579582015582749737956, −2.70785586088951357545083744290, 0,
2.70785586088951357545083744290, 3.63280981579582015582749737956, 4.45121417313434353743955773395, 5.02634003713131708178759643517, 6.65750338714462598945729693110, 6.95467925429763141050855852171, 8.266687228494963366525791727658, 8.936466461282421859104530205200, 10.31892881686011013076334032188
