L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s − 2·7-s + 3·8-s + 9-s − 2·10-s − 11-s + 12-s + 13-s + 2·14-s − 2·15-s − 16-s + 2·17-s − 18-s + 6·19-s − 2·20-s + 2·21-s + 22-s − 23-s − 3·24-s − 25-s − 26-s − 27-s + 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s + 0.436·21-s + 0.213·22-s − 0.208·23-s − 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.000609117\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.000609117\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.62106472728305922514031445513, −7.12362753946334775872436192264, −6.22730519707777300029588682650, −5.57578809947527492406207109498, −5.21778459596272272022795964140, −4.18326722206152054829905958328, −3.46328190649881193575585041173, −2.41890058866949169163793616203, −1.39715536453312817656775618277, −0.60016441256097470346533413647,
0.60016441256097470346533413647, 1.39715536453312817656775618277, 2.41890058866949169163793616203, 3.46328190649881193575585041173, 4.18326722206152054829905958328, 5.21778459596272272022795964140, 5.57578809947527492406207109498, 6.22730519707777300029588682650, 7.12362753946334775872436192264, 7.62106472728305922514031445513