| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 2·13-s − 15-s + 4·19-s + 21-s + 23-s + 25-s − 27-s + 4·31-s − 35-s + 2·37-s − 2·39-s + 12·41-s + 4·43-s + 45-s − 12·47-s + 49-s − 4·57-s − 12·59-s + 14·61-s − 63-s + 2·65-s + 4·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.320·39-s + 1.87·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.125·63-s + 0.248·65-s + 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260191986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260191986\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.74470138409163, −14.25100382794982, −13.80109199499849, −13.12396325639588, −12.83203210575365, −12.26969995716338, −11.54948471958486, −11.24659893063905, −10.63552100635406, −10.07440641346624, −9.489018797657130, −9.232894822008397, −8.331851797160352, −7.869676484755227, −7.118759588543026, −6.639067339653344, −5.988561661442570, −5.662896354306327, −4.904632996709151, −4.354708281065519, −3.558469409625235, −2.941368370835578, −2.168682648968459, −1.251361259086399, −0.6354819882324166,
0.6354819882324166, 1.251361259086399, 2.168682648968459, 2.941368370835578, 3.558469409625235, 4.354708281065519, 4.904632996709151, 5.662896354306327, 5.988561661442570, 6.639067339653344, 7.118759588543026, 7.869676484755227, 8.331851797160352, 9.232894822008397, 9.489018797657130, 10.07440641346624, 10.63552100635406, 11.24659893063905, 11.54948471958486, 12.26969995716338, 12.83203210575365, 13.12396325639588, 13.80109199499849, 14.25100382794982, 14.74470138409163
