| L(s) = 1 | + 10·7-s + 46·11-s − 34·13-s − 66·17-s + 104·19-s − 164·23-s − 224·29-s − 72·31-s − 22·37-s − 194·41-s + 108·43-s + 480·47-s − 243·49-s − 286·53-s − 426·59-s + 698·61-s + 328·67-s − 188·71-s − 740·73-s + 460·77-s + 1.16e3·79-s − 412·83-s − 1.20e3·89-s − 340·91-s − 1.38e3·97-s + 1.12e3·101-s − 758·103-s + ⋯ |
| L(s) = 1 | + 0.539·7-s + 1.26·11-s − 0.725·13-s − 0.941·17-s + 1.25·19-s − 1.48·23-s − 1.43·29-s − 0.417·31-s − 0.0977·37-s − 0.738·41-s + 0.383·43-s + 1.48·47-s − 0.708·49-s − 0.741·53-s − 0.940·59-s + 1.46·61-s + 0.598·67-s − 0.314·71-s − 1.18·73-s + 0.680·77-s + 1.66·79-s − 0.544·83-s − 1.43·89-s − 0.391·91-s − 1.44·97-s + 1.11·101-s − 0.725·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 46 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 164 T + p^{3} T^{2} \) |
| 29 | \( 1 + 224 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 22 T + p^{3} T^{2} \) |
| 41 | \( 1 + 194 T + p^{3} T^{2} \) |
| 43 | \( 1 - 108 T + p^{3} T^{2} \) |
| 47 | \( 1 - 480 T + p^{3} T^{2} \) |
| 53 | \( 1 + 286 T + p^{3} T^{2} \) |
| 59 | \( 1 + 426 T + p^{3} T^{2} \) |
| 61 | \( 1 - 698 T + p^{3} T^{2} \) |
| 67 | \( 1 - 328 T + p^{3} T^{2} \) |
| 71 | \( 1 + 188 T + p^{3} T^{2} \) |
| 73 | \( 1 + 740 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 412 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1206 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1384 T + p^{3} T^{2} \) |
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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
−8.574259556317163784698141509614, −7.65002321887676829533358310696, −7.04156272848248849584558376047, −6.09743328249542248670045640864, −5.26480338295921929394123047351, −4.31341305928207759413358955265, −3.58706692950687204880715250725, −2.25952018678081181530397701399, −1.39813602893720354901435763939, 0,
1.39813602893720354901435763939, 2.25952018678081181530397701399, 3.58706692950687204880715250725, 4.31341305928207759413358955265, 5.26480338295921929394123047351, 6.09743328249542248670045640864, 7.04156272848248849584558376047, 7.65002321887676829533358310696, 8.574259556317163784698141509614
