| L(s) = 1 | + 3-s − 7-s + 9-s + 3·11-s − 13-s + 7·17-s + 4·19-s − 21-s + 9·23-s + 27-s + 8·29-s − 4·31-s + 3·33-s + 3·37-s − 39-s − 5·41-s + 2·47-s − 6·49-s + 7·51-s + 3·53-s + 4·57-s + 12·59-s − 15·61-s − 63-s − 12·67-s + 9·69-s + 15·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 1.69·17-s + 0.917·19-s − 0.218·21-s + 1.87·23-s + 0.192·27-s + 1.48·29-s − 0.718·31-s + 0.522·33-s + 0.493·37-s − 0.160·39-s − 0.780·41-s + 0.291·47-s − 6/7·49-s + 0.980·51-s + 0.412·53-s + 0.529·57-s + 1.56·59-s − 1.92·61-s − 0.125·63-s − 1.46·67-s + 1.08·69-s + 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.518251113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.518251113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.15057970022349, −15.24571129231337, −14.90874874255460, −14.28390805791021, −13.94778419649031, −13.23706443661794, −12.67126741102077, −12.08699521202885, −11.67441124737187, −10.88596494059348, −10.15570389900707, −9.731724321358899, −9.141378823429718, −8.672683810863195, −7.856731615083818, −7.339009905851648, −6.773503707844457, −6.088269638466360, −5.238895873346774, −4.742861322066966, −3.718986663233474, −3.239120239860574, −2.676652073714998, −1.447012692165350, −0.8840017279395159,
0.8840017279395159, 1.447012692165350, 2.676652073714998, 3.239120239860574, 3.718986663233474, 4.742861322066966, 5.238895873346774, 6.088269638466360, 6.773503707844457, 7.339009905851648, 7.856731615083818, 8.672683810863195, 9.141378823429718, 9.731724321358899, 10.15570389900707, 10.88596494059348, 11.67441124737187, 12.08699521202885, 12.67126741102077, 13.23706443661794, 13.94778419649031, 14.28390805791021, 14.90874874255460, 15.24571129231337, 16.15057970022349
