L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 12-s + 2·13-s + 2·14-s − 15-s + 16-s − 6·17-s − 18-s − 8·19-s + 20-s + 2·21-s + 6·23-s + 24-s + 25-s − 2·26-s − 27-s − 2·28-s + 6·29-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 − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.436·21-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−14.76216469069701, −14.14702220389312, −13.49926140221966, −12.96732330941583, −12.67419885271692, −12.20323454796583, −11.28374245835801, −10.98037050452373, −10.61836931040775, −10.13129916881447, −9.410651919935735, −8.942717799889114, −8.670904337315159, −7.949359465296235, −7.120308925479816, −6.583046813010524, −6.449917199757965, −5.836603408570568, −5.022215307281986, −4.482480059574593, −3.755694886341706, −2.993473771478233, −2.273706936281613, −1.705473385094389, −0.7518947339911400, 0,
0.7518947339911400, 1.705473385094389, 2.273706936281613, 2.993473771478233, 3.755694886341706, 4.482480059574593, 5.022215307281986, 5.836603408570568, 6.449917199757965, 6.583046813010524, 7.120308925479816, 7.949359465296235, 8.670904337315159, 8.942717799889114, 9.410651919935735, 10.13129916881447, 10.61836931040775, 10.98037050452373, 11.28374245835801, 12.20323454796583, 12.67419885271692, 12.96732330941583, 13.49926140221966, 14.14702220389312, 14.76216469069701