| L(s) = 1 | + 9·5-s + 7-s − 63·11-s − 28·13-s + 72·17-s − 98·19-s − 126·23-s − 44·25-s − 126·29-s + 259·31-s + 9·35-s + 386·37-s − 450·41-s + 34·43-s + 54·47-s − 342·49-s − 693·53-s − 567·55-s − 180·59-s − 280·61-s − 252·65-s + 586·67-s − 504·71-s + 161·73-s − 63·77-s − 440·79-s − 999·83-s + ⋯ |
| L(s) = 1 | + 0.804·5-s + 0.0539·7-s − 1.72·11-s − 0.597·13-s + 1.02·17-s − 1.18·19-s − 1.14·23-s − 0.351·25-s − 0.806·29-s + 1.50·31-s + 0.0434·35-s + 1.71·37-s − 1.71·41-s + 0.120·43-s + 0.167·47-s − 0.997·49-s − 1.79·53-s − 1.39·55-s − 0.397·59-s − 0.587·61-s − 0.480·65-s + 1.06·67-s − 0.842·71-s + 0.258·73-s − 0.0932·77-s − 0.626·79-s − 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 63 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 + 98 T + p^{3} T^{2} \) |
| 23 | \( 1 + 126 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 259 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 450 T + p^{3} T^{2} \) |
| 43 | \( 1 - 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 54 T + p^{3} T^{2} \) |
| 53 | \( 1 + 693 T + p^{3} T^{2} \) |
| 59 | \( 1 + 180 T + p^{3} T^{2} \) |
| 61 | \( 1 + 280 T + p^{3} T^{2} \) |
| 67 | \( 1 - 586 T + p^{3} T^{2} \) |
| 71 | \( 1 + 504 T + p^{3} T^{2} \) |
| 73 | \( 1 - 161 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 999 T + p^{3} T^{2} \) |
| 89 | \( 1 - 882 T + p^{3} T^{2} \) |
| 97 | \( 1 + 721 T + p^{3} T^{2} \) |
| 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
−10.05032977169968184120798643203, −9.765804809381818496538718389241, −8.244308320272959837545716793848, −7.72724390411454406747907006507, −6.32654389860995327334353809725, −5.53215057220233203753126873725, −4.54213450159882338078877835495, −2.92285310453064962496669902575, −1.91461675808719239006423681821, 0,
1.91461675808719239006423681821, 2.92285310453064962496669902575, 4.54213450159882338078877835495, 5.53215057220233203753126873725, 6.32654389860995327334353809725, 7.72724390411454406747907006507, 8.244308320272959837545716793848, 9.765804809381818496538718389241, 10.05032977169968184120798643203
