L(s) = 1 | − 3-s − 7-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 21-s + 8·23-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 2·37-s + 2·39-s + 2·41-s − 12·43-s + 8·47-s + 49-s + 2·51-s + 6·53-s − 4·57-s − 4·59-s + 2·61-s − 63-s + 12·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s + 1.46·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.22970623553413, −15.01385939584636, −14.13406280315785, −13.55923828069938, −13.02834120621349, −12.74732959858203, −12.14308079113538, −11.48372327899064, −10.93485193373410, −10.65294786486582, −9.933799654818406, −9.354821304934509, −9.017390297648957, −8.059940443153103, −7.629083395411812, −6.861993280901185, −6.766996285682923, −5.516779965800145, −5.388114981750604, −4.897282613421039, −3.927772768792639, −3.326096328712812, −2.580441807160393, −1.889820560747848, −0.8140051888799191, 0,
0.8140051888799191, 1.889820560747848, 2.580441807160393, 3.326096328712812, 3.927772768792639, 4.897282613421039, 5.388114981750604, 5.516779965800145, 6.766996285682923, 6.861993280901185, 7.629083395411812, 8.059940443153103, 9.017390297648957, 9.354821304934509, 9.933799654818406, 10.65294786486582, 10.93485193373410, 11.48372327899064, 12.14308079113538, 12.74732959858203, 13.02834120621349, 13.55923828069938, 14.13406280315785, 15.01385939584636, 15.22970623553413