| L(s) = 1 | − 3·3-s − 14.5·5-s + 9·9-s + 10.1·11-s + 0.623·13-s + 43.5·15-s + 16.8·17-s − 75.7·19-s + 46.7·23-s + 85.5·25-s − 27·27-s + 73.5·29-s + 135.·31-s − 30.3·33-s + 133.·37-s − 1.87·39-s + 69.3·41-s + 279.·43-s − 130.·45-s − 372.·47-s − 50.4·51-s + 656.·53-s − 146.·55-s + 227.·57-s + 730.·59-s − 39.5·61-s − 9.05·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.29·5-s + 0.333·9-s + 0.277·11-s + 0.0133·13-s + 0.749·15-s + 0.239·17-s − 0.915·19-s + 0.423·23-s + 0.684·25-s − 0.192·27-s + 0.470·29-s + 0.784·31-s − 0.160·33-s + 0.591·37-s − 0.00768·39-s + 0.263·41-s + 0.990·43-s − 0.432·45-s − 1.15·47-s − 0.138·51-s + 1.70·53-s − 0.359·55-s + 0.528·57-s + 1.61·59-s − 0.0829·61-s − 0.0172·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 14.5T + 125T^{2} \) |
| 11 | \( 1 - 10.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.623T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 73.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 69.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 279.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 372.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 656.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 730.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 39.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 417.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 831.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.19e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 678.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.65e3T + 9.12e5T^{2} \) |
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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
−8.828442746174199415525385839245, −8.147531351919606811817165556910, −7.30072673562718736942980585146, −6.57400593537917437694237385984, −5.58755090346612643593921234088, −4.47146115771470077234563565350, −3.93956394909069476051115920787, −2.70561639700481847680246692637, −1.09364473732739488342367524078, 0,
1.09364473732739488342367524078, 2.70561639700481847680246692637, 3.93956394909069476051115920787, 4.47146115771470077234563565350, 5.58755090346612643593921234088, 6.57400593537917437694237385984, 7.30072673562718736942980585146, 8.147531351919606811817165556910, 8.828442746174199415525385839245
