| L(s) = 1 | + 3·3-s − 8·4-s − 17·7-s + 9·9-s + 39·11-s − 24·12-s − 13·13-s + 64·16-s + 93·17-s − 76·19-s − 51·21-s − 174·23-s + 27·27-s + 136·28-s − 57·29-s + 281·31-s + 117·33-s − 72·36-s + 286·37-s − 39·39-s − 264·41-s + 166·43-s − 312·44-s − 315·47-s + 192·48-s − 54·49-s + 279·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.917·7-s + 1/3·9-s + 1.06·11-s − 0.577·12-s − 0.277·13-s + 16-s + 1.32·17-s − 0.917·19-s − 0.529·21-s − 1.57·23-s + 0.192·27-s + 0.917·28-s − 0.364·29-s + 1.62·31-s + 0.617·33-s − 1/3·36-s + 1.27·37-s − 0.160·39-s − 1.00·41-s + 0.588·43-s − 1.06·44-s − 0.977·47-s + 0.577·48-s − 0.157·49-s + 0.766·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 17 T + p^{3} T^{2} \) |
| 11 | \( 1 - 39 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 174 T + p^{3} T^{2} \) |
| 29 | \( 1 + 57 T + p^{3} T^{2} \) |
| 31 | \( 1 - 281 T + p^{3} T^{2} \) |
| 37 | \( 1 - 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 264 T + p^{3} T^{2} \) |
| 43 | \( 1 - 166 T + p^{3} T^{2} \) |
| 47 | \( 1 + 315 T + p^{3} T^{2} \) |
| 53 | \( 1 - 207 T + p^{3} T^{2} \) |
| 59 | \( 1 + 519 T + p^{3} T^{2} \) |
| 61 | \( 1 + 511 T + p^{3} T^{2} \) |
| 67 | \( 1 + 803 T + p^{3} T^{2} \) |
| 71 | \( 1 - 624 T + p^{3} T^{2} \) |
| 73 | \( 1 + 698 T + p^{3} T^{2} \) |
| 79 | \( 1 - 554 T + p^{3} T^{2} \) |
| 83 | \( 1 + 507 T + p^{3} T^{2} \) |
| 89 | \( 1 - 360 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1370 T + p^{3} T^{2} \) |
| 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
−9.427512004996867925232007516288, −8.373988593603319473569460336610, −7.81184451559920717634767827891, −6.54711496532145626640403816760, −5.85133443027087436188735004852, −4.50679771782056982835379097950, −3.83341840698490577214809546249, −2.91888347185253102209988346062, −1.36433408587650242298184215668, 0,
1.36433408587650242298184215668, 2.91888347185253102209988346062, 3.83341840698490577214809546249, 4.50679771782056982835379097950, 5.85133443027087436188735004852, 6.54711496532145626640403816760, 7.81184451559920717634767827891, 8.373988593603319473569460336610, 9.427512004996867925232007516288
