| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s + 2·13-s − 15-s + 6·17-s − 19-s − 4·23-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s + 10·37-s − 2·39-s + 6·41-s + 45-s − 12·47-s − 7·49-s − 6·51-s + 10·53-s + 4·55-s + 57-s − 10·61-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.149·45-s − 1.75·47-s − 49-s − 0.840·51-s + 1.37·53-s + 0.539·55-s + 0.132·57-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827873442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827873442\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.287180121684984784965900213765, −8.147138876990079866417523759053, −7.50281712761657102889086692836, −6.35198931281804077801342630786, −6.08317621586618125737703438550, −5.16940849835265890817106579659, −4.13362534201749987180283134496, −3.39156409204262818849202841222, −1.94416852136943820176158401884, −0.962041023238845099053363878952,
0.962041023238845099053363878952, 1.94416852136943820176158401884, 3.39156409204262818849202841222, 4.13362534201749987180283134496, 5.16940849835265890817106579659, 6.08317621586618125737703438550, 6.35198931281804077801342630786, 7.50281712761657102889086692836, 8.147138876990079866417523759053, 9.287180121684984784965900213765
