| L(s) = 1 | − 2.19·2-s + 2.83·4-s + 0.364·5-s + 3.03·7-s − 1.83·8-s − 0.801·10-s − 3.36·11-s − 4.66·13-s − 6.66·14-s − 1.63·16-s − 7.83·17-s + 3.46·19-s + 1.03·20-s + 7.39·22-s + 6.23·23-s − 4.86·25-s + 10.2·26-s + 8.59·28-s − 8.79·29-s + 31-s + 7.26·32-s + 17.2·34-s + 1.10·35-s + 1.86·37-s − 7.62·38-s − 0.668·40-s + 5.86·41-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.41·4-s + 0.162·5-s + 1.14·7-s − 0.648·8-s − 0.253·10-s − 1.01·11-s − 1.29·13-s − 1.78·14-s − 0.408·16-s − 1.90·17-s + 0.796·19-s + 0.230·20-s + 1.57·22-s + 1.29·23-s − 0.973·25-s + 2.01·26-s + 1.62·28-s − 1.63·29-s + 0.179·31-s + 1.28·32-s + 2.95·34-s + 0.186·35-s + 0.306·37-s − 1.23·38-s − 0.105·40-s + 0.916·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 5 | \( 1 - 0.364T + 5T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 + 3.36T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 17 | \( 1 + 7.83T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 8.79T + 29T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 8.93T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 - 4.46T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 - 0.177T + 79T^{2} \) |
| 83 | \( 1 + 8.33T + 83T^{2} \) |
| 89 | \( 1 - 8.73T + 89T^{2} \) |
| 97 | \( 1 - 7.33T + 97T^{2} \) |
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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
−9.541614166561680264564464962142, −9.107029799273595636647711239081, −8.011044114076593979000234140113, −7.61012443681847080959431505390, −6.77150987436833601208467121007, −5.30723597724125114556483716154, −4.55796904433071459449645841423, −2.58738001748622867373060222373, −1.73677759593141191973907963776, 0,
1.73677759593141191973907963776, 2.58738001748622867373060222373, 4.55796904433071459449645841423, 5.30723597724125114556483716154, 6.77150987436833601208467121007, 7.61012443681847080959431505390, 8.011044114076593979000234140113, 9.107029799273595636647711239081, 9.541614166561680264564464962142
