| L(s) = 1 | + 2·5-s − 7-s + 2·11-s + 4·13-s − 5·17-s − 6·19-s + 2·23-s + 3·25-s − 13·29-s + 5·31-s − 2·35-s − 3·37-s − 11·41-s + 4·43-s − 16·47-s − 9·49-s + 3·53-s + 4·55-s − 9·59-s + 2·61-s + 8·65-s − 67-s − 11·71-s + 4·73-s − 2·77-s + 4·79-s + 13·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.603·11-s + 1.10·13-s − 1.21·17-s − 1.37·19-s + 0.417·23-s + 3/5·25-s − 2.41·29-s + 0.898·31-s − 0.338·35-s − 0.493·37-s − 1.71·41-s + 0.609·43-s − 2.33·47-s − 9/7·49-s + 0.412·53-s + 0.539·55-s − 1.17·59-s + 0.256·61-s + 0.992·65-s − 0.122·67-s − 1.30·71-s + 0.468·73-s − 0.227·77-s + 0.450·79-s + 1.42·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68558400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 72 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 108 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 104 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 134 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 130 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 134 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 170 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56971372418466804539927384322, −7.21637256501280162113959343014, −6.66893558144063148510401773898, −6.63240962341754208822362135949, −6.25132032805455573608279391080, −6.15734314558101067821059669810, −5.51449684210610830824045835018, −5.31884386453976846140499991123, −4.72628109769820027393430279812, −4.58795126144026376387347860670, −3.98423927982216671634500167293, −3.71700512779088918920546717021, −3.30145101932707796468791180031, −2.95794496620772963152941574193, −2.19342976160115189162269008063, −2.10318712118214352601061277618, −1.38803074314310923480415015236, −1.35592565436354123882959875555, 0, 0,
1.35592565436354123882959875555, 1.38803074314310923480415015236, 2.10318712118214352601061277618, 2.19342976160115189162269008063, 2.95794496620772963152941574193, 3.30145101932707796468791180031, 3.71700512779088918920546717021, 3.98423927982216671634500167293, 4.58795126144026376387347860670, 4.72628109769820027393430279812, 5.31884386453976846140499991123, 5.51449684210610830824045835018, 6.15734314558101067821059669810, 6.25132032805455573608279391080, 6.63240962341754208822362135949, 6.66893558144063148510401773898, 7.21637256501280162113959343014, 7.56971372418466804539927384322
