| L(s) = 1 | − 2·5-s + 10·11-s + 14·19-s − 25-s − 10·29-s − 6·31-s − 14·41-s + 10·49-s − 20·55-s − 30·59-s + 28·61-s + 10·71-s + 16·79-s + 6·89-s − 28·95-s + 30·101-s + 18·109-s + 53·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + 12·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 3.01·11-s + 3.21·19-s − 1/5·25-s − 1.85·29-s − 1.07·31-s − 2.18·41-s + 10/7·49-s − 2.69·55-s − 3.90·59-s + 3.58·61-s + 1.18·71-s + 1.80·79-s + 0.635·89-s − 2.87·95-s + 2.98·101-s + 1.72·109-s + 4.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + 0.963·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997927856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997927856\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.751247200358200256677502074403, −8.739766977327316168551549030044, −7.973103881985362746939187645620, −7.59925169840168080336403985486, −7.25816312128100823202082167354, −7.23408295871254035710001612488, −6.56678118655913522222147313566, −6.37484696475546100397345377836, −5.64194709881183865967712858245, −5.61749562875058607116179341784, −4.82508550326625055024023385626, −4.75515824880593334908041869324, −3.74976450356557121483933309561, −3.72212784264987148652800031930, −3.59387681436439858374163140986, −3.14899033437529600440979433568, −2.10778927255725965753624548073, −1.68918136824575440729859233700, −1.15113616788828396308145435499, −0.61838830697776382225402053268,
0.61838830697776382225402053268, 1.15113616788828396308145435499, 1.68918136824575440729859233700, 2.10778927255725965753624548073, 3.14899033437529600440979433568, 3.59387681436439858374163140986, 3.72212784264987148652800031930, 3.74976450356557121483933309561, 4.75515824880593334908041869324, 4.82508550326625055024023385626, 5.61749562875058607116179341784, 5.64194709881183865967712858245, 6.37484696475546100397345377836, 6.56678118655913522222147313566, 7.23408295871254035710001612488, 7.25816312128100823202082167354, 7.59925169840168080336403985486, 7.973103881985362746939187645620, 8.739766977327316168551549030044, 8.751247200358200256677502074403
