L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 2·13-s + 14-s + 16-s + 3.65·17-s + 2.82·19-s + 20-s − 22-s − 2.82·23-s + 25-s − 2·26-s − 28-s − 8.82·29-s + 5.65·31-s − 32-s − 3.65·34-s − 35-s + 8.82·37-s − 2.82·38-s − 40-s − 4.82·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 0.250·16-s + 0.886·17-s + 0.648·19-s + 0.223·20-s − 0.213·22-s − 0.589·23-s + 0.200·25-s − 0.392·26-s − 0.188·28-s − 1.63·29-s + 1.01·31-s − 0.176·32-s − 0.627·34-s − 0.169·35-s + 1.45·37-s − 0.458·38-s − 0.158·40-s − 0.754·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.627463752\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.627463752\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 8.82T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−7.958396207286525661007157354510, −7.37182532142529838037535128381, −6.59960604441439169607713175404, −5.89395603376242765595164866473, −5.44264740363302967203712321731, −4.23329283308185361339478924468, −3.45211045655002986373715699057, −2.62640663004094932202407881308, −1.64138778229613581179116197128, −0.75523982962365721079831247592,
0.75523982962365721079831247592, 1.64138778229613581179116197128, 2.62640663004094932202407881308, 3.45211045655002986373715699057, 4.23329283308185361339478924468, 5.44264740363302967203712321731, 5.89395603376242765595164866473, 6.59960604441439169607713175404, 7.37182532142529838037535128381, 7.958396207286525661007157354510