| L(s) = 1 | + 81.4·5-s + 49·7-s + 340.·11-s − 1.10e3·13-s + 197.·17-s − 2.33e3·19-s − 2.60e3·23-s + 3.50e3·25-s − 7.91e3·29-s − 9.04e3·31-s + 3.98e3·35-s − 5.47e3·37-s + 1.52e4·41-s − 3.82e3·43-s − 1.94e3·47-s + 2.40e3·49-s − 2.62e4·53-s + 2.77e4·55-s + 4.58e4·59-s − 4.34e4·61-s − 8.96e4·65-s + 1.58e4·67-s + 2.41e4·71-s + 6.90e4·73-s + 1.66e4·77-s + 6.09e4·79-s − 5.22e4·83-s + ⋯ |
| L(s) = 1 | + 1.45·5-s + 0.377·7-s + 0.847·11-s − 1.80·13-s + 0.165·17-s − 1.48·19-s − 1.02·23-s + 1.12·25-s − 1.74·29-s − 1.69·31-s + 0.550·35-s − 0.657·37-s + 1.41·41-s − 0.315·43-s − 0.128·47-s + 0.142·49-s − 1.28·53-s + 1.23·55-s + 1.71·59-s − 1.49·61-s − 2.63·65-s + 0.431·67-s + 0.567·71-s + 1.51·73-s + 0.320·77-s + 1.09·79-s − 0.832·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 5 | \( 1 - 81.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 340.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.10e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 197.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.82e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.62e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.90e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.49e4T + 8.58e9T^{2} \) |
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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
−9.567767178512870587682137245733, −9.131036245862864957765690512170, −7.81985531457090474611893169462, −6.83922864900154506249809388919, −5.90034977237662648862442148715, −5.09822234484879880557327616876, −3.93240924243128955304812489077, −2.26108204243016073910759863426, −1.77469360753255666603762645130, 0,
1.77469360753255666603762645130, 2.26108204243016073910759863426, 3.93240924243128955304812489077, 5.09822234484879880557327616876, 5.90034977237662648862442148715, 6.83922864900154506249809388919, 7.81985531457090474611893169462, 9.131036245862864957765690512170, 9.567767178512870587682137245733
