| L(s) = 1 | + 4·7-s + 16·19-s + 10·25-s − 8·31-s − 20·37-s + 9·49-s + 40·103-s − 4·109-s + 22·121-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 40·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 3.67·19-s + 2·25-s − 1.43·31-s − 3.28·37-s + 9/7·49-s + 3.94·103-s − 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-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 − 1.69·169-s + 0.0760·173-s + 3.02·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.458728940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.458728940\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.596725383686562905316882366316, −8.420978027843578433916183539214, −7.64252170234416689478154199921, −7.59421999213631114320567289337, −7.13726396288468678614656558958, −7.12258079671961120946595011882, −6.51649027903511166834528359370, −5.88548840932524991046662378062, −5.37836403329105250998514994545, −5.33179874416258980053124884973, −4.90782569453751539991438154473, −4.76758534558069341909514312899, −4.01673459654689862047638011310, −3.49980767992939549339499021988, −3.14475856152786744212207588476, −2.97540378446622941865507341481, −1.88976637909954471463017660692, −1.82893367938793672962508808326, −1.12604933707791460641359255989, −0.69351545017976367689237117372,
0.69351545017976367689237117372, 1.12604933707791460641359255989, 1.82893367938793672962508808326, 1.88976637909954471463017660692, 2.97540378446622941865507341481, 3.14475856152786744212207588476, 3.49980767992939549339499021988, 4.01673459654689862047638011310, 4.76758534558069341909514312899, 4.90782569453751539991438154473, 5.33179874416258980053124884973, 5.37836403329105250998514994545, 5.88548840932524991046662378062, 6.51649027903511166834528359370, 7.12258079671961120946595011882, 7.13726396288468678614656558958, 7.59421999213631114320567289337, 7.64252170234416689478154199921, 8.420978027843578433916183539214, 8.596725383686562905316882366316
