| L(s) = 1 | + 4·13-s − 2·17-s + 8·19-s + 10·25-s + 8·43-s − 16·47-s − 2·49-s − 12·53-s − 24·59-s − 24·67-s + 24·83-s + 20·89-s + 12·101-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 0.485·17-s + 1.83·19-s + 2·25-s + 1.21·43-s − 2.33·47-s − 2/7·49-s − 1.64·53-s − 3.12·59-s − 2.93·67-s + 2.63·83-s + 2.11·89-s + 1.19·101-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.585255442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.585255442\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.225287709285959191736619465834, −8.754876984127810087531641499762, −8.478535596709911215007179621237, −7.78987309809769759559479969485, −7.70263282682119755145400937233, −7.36375332958445321348303956078, −6.73094065576395437782977227829, −6.33860563095576610389997167655, −6.16631158193617952287891358636, −5.73637868312127189236441813779, −4.96710679513710451000328063963, −4.78669127463215302246567304886, −4.62365877530828677142891695210, −3.73200549467605985910475138340, −3.21488755984848636972305576028, −3.19891141047355361785014440707, −2.55317958112811488980896090028, −1.57571465254573096698888981131, −1.41278891630442857583352756514, −0.57352570508392388865017562551,
0.57352570508392388865017562551, 1.41278891630442857583352756514, 1.57571465254573096698888981131, 2.55317958112811488980896090028, 3.19891141047355361785014440707, 3.21488755984848636972305576028, 3.73200549467605985910475138340, 4.62365877530828677142891695210, 4.78669127463215302246567304886, 4.96710679513710451000328063963, 5.73637868312127189236441813779, 6.16631158193617952287891358636, 6.33860563095576610389997167655, 6.73094065576395437782977227829, 7.36375332958445321348303956078, 7.70263282682119755145400937233, 7.78987309809769759559479969485, 8.478535596709911215007179621237, 8.754876984127810087531641499762, 9.225287709285959191736619465834
