| L(s) = 1 | + 2·7-s + 4·13-s + 4·19-s + 4·31-s − 2·37-s + 10·43-s + 3·49-s + 16·61-s − 14·67-s + 16·73-s + 10·79-s + 8·91-s − 20·97-s + 4·103-s − 2·109-s + 5·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.10·13-s + 0.917·19-s + 0.718·31-s − 0.328·37-s + 1.52·43-s + 3/7·49-s + 2.04·61-s − 1.71·67-s + 1.87·73-s + 1.12·79-s + 0.838·91-s − 2.03·97-s + 0.394·103-s − 0.191·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-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 − 1.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.544243890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.544243890\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.113444027558810324798777966243, −7.979636124044077837789074450258, −7.51106661163631629379564321726, −7.25891588786795890978290659647, −6.67439656143676099955168987087, −6.59788041369182469264473123034, −6.06386182131145744565369306558, −5.61554044775219584328992664360, −5.44219256625498784791767463257, −5.08966483996453124771926940109, −4.42778807635828690546122299178, −4.38910803295400032057045263275, −3.66787753000870042916876877309, −3.63301223888656291973975169452, −2.86928509294642531952958975129, −2.66451721083818135937228720008, −1.95560350618348496187167926898, −1.62931467179059793975593563183, −0.943743877056097002824036133708, −0.66434007814031727085461619397,
0.66434007814031727085461619397, 0.943743877056097002824036133708, 1.62931467179059793975593563183, 1.95560350618348496187167926898, 2.66451721083818135937228720008, 2.86928509294642531952958975129, 3.63301223888656291973975169452, 3.66787753000870042916876877309, 4.38910803295400032057045263275, 4.42778807635828690546122299178, 5.08966483996453124771926940109, 5.44219256625498784791767463257, 5.61554044775219584328992664360, 6.06386182131145744565369306558, 6.59788041369182469264473123034, 6.67439656143676099955168987087, 7.25891588786795890978290659647, 7.51106661163631629379564321726, 7.979636124044077837789074450258, 8.113444027558810324798777966243
