| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 6·13-s + 15-s − 17-s + 4·19-s − 21-s + 25-s + 27-s + 2·29-s − 35-s + 6·37-s + 6·39-s − 10·41-s + 4·43-s + 45-s − 4·47-s + 49-s − 51-s − 6·53-s + 4·57-s + 4·59-s + 14·61-s − 63-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s − 0.242·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.169·35-s + 0.986·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.140·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s + 1.79·61-s − 0.125·63-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.214636297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.214636297\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.40912594126633, −13.26259021755040, −13.00376166196731, −11.99481924384025, −11.84776033687885, −11.10209773146724, −10.65437091289177, −10.17670926517198, −9.617701872034077, −9.188743943993400, −8.757688539709872, −8.168108832245588, −7.850843852614719, −6.987889929987741, −6.681824178236751, −6.048144305990408, −5.641484848063541, −4.959236867139111, −4.287795527629924, −3.722863638958981, −3.167729102752026, −2.744691547504404, −1.860559150354531, −1.346814427304953, −0.6307877917505678,
0.6307877917505678, 1.346814427304953, 1.860559150354531, 2.744691547504404, 3.167729102752026, 3.722863638958981, 4.287795527629924, 4.959236867139111, 5.641484848063541, 6.048144305990408, 6.681824178236751, 6.987889929987741, 7.850843852614719, 8.168108832245588, 8.757688539709872, 9.188743943993400, 9.617701872034077, 10.17670926517198, 10.65437091289177, 11.10209773146724, 11.84776033687885, 11.99481924384025, 13.00376166196731, 13.26259021755040, 13.40912594126633
