L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 12-s − 13-s − 15-s + 16-s + 6·17-s + 18-s − 20-s − 4·23-s + 24-s + 25-s − 26-s + 27-s − 30-s + 32-s + 6·34-s + 36-s + 6·37-s − 39-s − 40-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.182·30-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.986·37-s − 0.160·39-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.86657336675851, −12.33439895854158, −12.07606289089365, −11.65643403968043, −10.95855112121713, −10.74270905226299, −10.04019433871408, −9.578600541275313, −9.408729248989445, −8.509535726071079, −8.091739435961584, −7.768705937604711, −7.409922298586990, −6.731901844613836, −6.292732289434186, −5.798166794053443, −5.131046715257336, −4.856902975712537, −4.085778898464473, −3.792623524563603, −3.260794353244919, −2.743203904261683, −2.230811694417755, −1.499116489428893, −0.9521092752723906, 0,
0.9521092752723906, 1.499116489428893, 2.230811694417755, 2.743203904261683, 3.260794353244919, 3.792623524563603, 4.085778898464473, 4.856902975712537, 5.131046715257336, 5.798166794053443, 6.292732289434186, 6.731901844613836, 7.409922298586990, 7.768705937604711, 8.091739435961584, 8.509535726071079, 9.408729248989445, 9.578600541275313, 10.04019433871408, 10.74270905226299, 10.95855112121713, 11.65643403968043, 12.07606289089365, 12.33439895854158, 12.86657336675851