| L(s) = 1 | + 22·7-s + 14·11-s − 30·13-s − 62·17-s − 120·19-s − 188·23-s − 96·29-s + 184·31-s + 406·37-s − 130·41-s + 148·43-s − 448·47-s + 141·49-s + 414·53-s − 266·59-s − 838·61-s + 248·67-s − 1.02e3·71-s + 484·73-s + 308·77-s − 48·79-s − 548·83-s + 650·89-s − 660·91-s − 1.81e3·97-s − 1.68e3·101-s − 298·103-s + ⋯ |
| L(s) = 1 | + 1.18·7-s + 0.383·11-s − 0.640·13-s − 0.884·17-s − 1.44·19-s − 1.70·23-s − 0.614·29-s + 1.06·31-s + 1.80·37-s − 0.495·41-s + 0.524·43-s − 1.39·47-s + 0.411·49-s + 1.07·53-s − 0.586·59-s − 1.75·61-s + 0.452·67-s − 1.70·71-s + 0.775·73-s + 0.455·77-s − 0.0683·79-s − 0.724·83-s + 0.774·89-s − 0.760·91-s − 1.90·97-s − 1.66·101-s − 0.285·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 19 | \( 1 + 120 T + p^{3} T^{2} \) |
| 23 | \( 1 + 188 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 - 184 T + p^{3} T^{2} \) |
| 37 | \( 1 - 406 T + p^{3} T^{2} \) |
| 41 | \( 1 + 130 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 448 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 266 T + p^{3} T^{2} \) |
| 61 | \( 1 + 838 T + p^{3} T^{2} \) |
| 67 | \( 1 - 248 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 - 484 T + p^{3} T^{2} \) |
| 79 | \( 1 + 48 T + p^{3} T^{2} \) |
| 83 | \( 1 + 548 T + p^{3} T^{2} \) |
| 89 | \( 1 - 650 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1816 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.279607104843375127706806367958, −8.311736152448270502546489978889, −7.82856448290452422805201727195, −6.67454368382943233492261695619, −5.87053498555330577616363415029, −4.61368466106746130303195663871, −4.17734777298160475681840973314, −2.50096518410165730966077872902, −1.62772360529634250944334349333, 0,
1.62772360529634250944334349333, 2.50096518410165730966077872902, 4.17734777298160475681840973314, 4.61368466106746130303195663871, 5.87053498555330577616363415029, 6.67454368382943233492261695619, 7.82856448290452422805201727195, 8.311736152448270502546489978889, 9.279607104843375127706806367958
