| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 11-s − 12-s + 13-s + 4·14-s + 16-s + 17-s − 18-s − 4·19-s + 4·21-s + 22-s + 24-s − 26-s − 27-s − 4·28-s − 6·29-s − 10·31-s − 32-s + 33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.872·21-s + 0.213·22-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.87366338414342, −12.47034704596265, −12.13755865198647, −11.50571781971197, −10.94745640938877, −10.78742838773532, −10.15732812630599, −9.876807543581066, −9.353864522875585, −8.952698898257411, −8.583644123457854, −7.809024969896208, −7.476281897545110, −6.880850835234573, −6.631995593333458, −5.944160277138153, −5.800321765807785, −5.161230189432081, −4.489786304833674, −3.786665192094638, −3.421407661225926, −2.952661534003598, −2.079271093754716, −1.773910651168864, −0.8755168482725797, 0, 0,
0.8755168482725797, 1.773910651168864, 2.079271093754716, 2.952661534003598, 3.421407661225926, 3.786665192094638, 4.489786304833674, 5.161230189432081, 5.800321765807785, 5.944160277138153, 6.631995593333458, 6.880850835234573, 7.476281897545110, 7.809024969896208, 8.583644123457854, 8.952698898257411, 9.353864522875585, 9.876807543581066, 10.15732812630599, 10.78742838773532, 10.94745640938877, 11.50571781971197, 12.13755865198647, 12.47034704596265, 12.87366338414342
