| L(s) = 1 | + 2·2-s + 4·4-s + 7·5-s + 7·7-s + 8·8-s + 14·10-s − 11·11-s − 67·13-s + 14·14-s + 16·16-s − 30·17-s − 7·19-s + 28·20-s − 22·22-s − 28·23-s − 76·25-s − 134·26-s + 28·28-s − 121·29-s − 310·31-s + 32·32-s − 60·34-s + 49·35-s − 71·37-s − 14·38-s + 56·40-s + 180·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.626·5-s + 0.377·7-s + 0.353·8-s + 0.442·10-s − 0.301·11-s − 1.42·13-s + 0.267·14-s + 1/4·16-s − 0.428·17-s − 0.0845·19-s + 0.313·20-s − 0.213·22-s − 0.253·23-s − 0.607·25-s − 1.01·26-s + 0.188·28-s − 0.774·29-s − 1.79·31-s + 0.176·32-s − 0.302·34-s + 0.236·35-s − 0.315·37-s − 0.0597·38-s + 0.221·40-s + 0.685·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 67 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 7 T + p^{3} T^{2} \) |
| 23 | \( 1 + 28 T + p^{3} T^{2} \) |
| 29 | \( 1 + 121 T + p^{3} T^{2} \) |
| 31 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 71 T + p^{3} T^{2} \) |
| 41 | \( 1 - 180 T + p^{3} T^{2} \) |
| 43 | \( 1 + 108 T + p^{3} T^{2} \) |
| 47 | \( 1 + 71 T + p^{3} T^{2} \) |
| 53 | \( 1 + 128 T + p^{3} T^{2} \) |
| 59 | \( 1 - 429 T + p^{3} T^{2} \) |
| 61 | \( 1 - 22 T + p^{3} T^{2} \) |
| 67 | \( 1 + 803 T + p^{3} T^{2} \) |
| 71 | \( 1 + 468 T + p^{3} T^{2} \) |
| 73 | \( 1 + 117 T + p^{3} T^{2} \) |
| 79 | \( 1 + 96 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1122 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 92 T + p^{3} T^{2} \) |
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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
−8.904838613736194506378769968733, −7.70061384693885111434857815193, −7.21726781719692113450559084611, −6.12271885235008361613748041409, −5.38216107645771569084856371447, −4.69457782279696940628856385650, −3.65563577989240869784434108319, −2.43489982006772064109432302748, −1.78459722927078568737247054476, 0,
1.78459722927078568737247054476, 2.43489982006772064109432302748, 3.65563577989240869784434108319, 4.69457782279696940628856385650, 5.38216107645771569084856371447, 6.12271885235008361613748041409, 7.21726781719692113450559084611, 7.70061384693885111434857815193, 8.904838613736194506378769968733
