L(s) = 1 | − 5-s + 2·19-s + 2·23-s + 25-s − 2·47-s − 49-s + 2·53-s − 2·95-s − 2·115-s + ⋯ |
L(s) = 1 | − 5-s + 2·19-s + 2·23-s + 25-s − 2·47-s − 49-s + 2·53-s − 2·95-s − 2·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9702192579\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9702192579\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.634931413357600399097750455912, −8.913000184119328056560084500575, −8.052979170327115532331260603797, −7.32017508356596966505606750782, −6.71283500152691600012809026495, −5.38023611699060471862691968533, −4.76464489215526970738269688408, −3.57428243889269261079699245640, −2.91462286210826165240138843168, −1.12511781657333200041863106708,
1.12511781657333200041863106708, 2.91462286210826165240138843168, 3.57428243889269261079699245640, 4.76464489215526970738269688408, 5.38023611699060471862691968533, 6.71283500152691600012809026495, 7.32017508356596966505606750782, 8.052979170327115532331260603797, 8.913000184119328056560084500575, 9.634931413357600399097750455912