| L(s) = 1 | + 5-s + 4·7-s + 4·13-s + 4·17-s + 8·19-s − 4·23-s + 25-s + 8·29-s − 4·31-s + 4·35-s + 6·37-s + 8·41-s + 4·43-s − 12·47-s + 9·49-s + 10·53-s + 8·61-s + 4·65-s + 8·67-s − 12·71-s + 12·73-s − 8·79-s − 4·83-s + 4·85-s − 10·89-s + 16·91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.10·13-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s + 0.676·35-s + 0.986·37-s + 1.24·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s + 1.37·53-s + 1.02·61-s + 0.496·65-s + 0.977·67-s − 1.42·71-s + 1.40·73-s − 0.900·79-s − 0.439·83-s + 0.433·85-s − 1.05·89-s + 1.67·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.300135319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.300135319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.67501143393989, −14.77162645625086, −14.39739820803464, −14.04817189257768, −13.53075981374586, −12.90392961370595, −12.09636678554797, −11.74196042487527, −11.14444452856937, −10.75695854911555, −9.862870269493750, −9.651939121294422, −8.723006197000180, −8.215972100939494, −7.803859343404783, −7.179534706142992, −6.358097477304605, −5.535075263505796, −5.424713303442933, −4.515602752870906, −3.893540170538027, −3.071304348013985, −2.291293559777405, −1.283497217694413, −1.047282473464941,
1.047282473464941, 1.283497217694413, 2.291293559777405, 3.071304348013985, 3.893540170538027, 4.515602752870906, 5.424713303442933, 5.535075263505796, 6.358097477304605, 7.179534706142992, 7.803859343404783, 8.215972100939494, 8.723006197000180, 9.651939121294422, 9.862870269493750, 10.75695854911555, 11.14444452856937, 11.74196042487527, 12.09636678554797, 12.90392961370595, 13.53075981374586, 14.04817189257768, 14.39739820803464, 14.77162645625086, 15.67501143393989
