L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 5·7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 5·14-s − 15-s + 16-s + 17-s − 18-s + 20-s + 5·21-s − 22-s + 24-s + 25-s − 26-s − 27-s − 5·28-s − 4·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.223·20-s + 1.09·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.944·28-s − 0.742·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.60040045132829, −13.04644235735941, −12.63085541410228, −12.35565692875097, −11.74210068012404, −11.20588786503245, −10.72669338892466, −10.25236802037752, −9.842588918638020, −9.417901189336656, −9.134470567970497, −8.522711551731598, −7.904273453494251, −7.212929704931915, −6.788862587823235, −6.518991916746689, −5.958781515377684, −5.514910259918786, −5.009558550877711, −4.016254913696140, −3.579581510285500, −3.114674452796539, −2.428838407118626, −1.676933295994371, −1.119870574719613, 0, 0,
1.119870574719613, 1.676933295994371, 2.428838407118626, 3.114674452796539, 3.579581510285500, 4.016254913696140, 5.009558550877711, 5.514910259918786, 5.958781515377684, 6.518991916746689, 6.788862587823235, 7.212929704931915, 7.904273453494251, 8.522711551731598, 9.134470567970497, 9.417901189336656, 9.842588918638020, 10.25236802037752, 10.72669338892466, 11.20588786503245, 11.74210068012404, 12.35565692875097, 12.63085541410228, 13.04644235735941, 13.60040045132829