| L(s) = 1 | + 4·11-s + 2·19-s − 8·29-s − 10·31-s + 12·41-s + 14·49-s + 10·61-s + 4·71-s + 22·79-s − 20·89-s − 24·101-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.458·19-s − 1.48·29-s − 1.79·31-s + 1.87·41-s + 2·49-s + 1.28·61-s + 0.474·71-s + 2.47·79-s − 2.11·89-s − 2.38·101-s − 1.72·109-s − 0.909·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.943184924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.943184924\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.225581177931808512771815524367, −7.976411683820534792778389981786, −7.67419059854830156854979116121, −7.21885384900375487185320826446, −6.84611206779404725206113332739, −6.78405464030845245417259376330, −6.21740941516161929256826489364, −5.60246499593008097503309344670, −5.46513242743369861100795806153, −5.44899571892131381918563815503, −4.56320977329341254096274747923, −4.16571129256246658456308924274, −3.87652428614993324478657608781, −3.70717818248219795825376849036, −3.03506026265434456505779216681, −2.62506064172105849044173494636, −2.00252257611571403326738883369, −1.72382265634785801797966192592, −1.02795872785757080851385569456, −0.51194152760306697267346069245,
0.51194152760306697267346069245, 1.02795872785757080851385569456, 1.72382265634785801797966192592, 2.00252257611571403326738883369, 2.62506064172105849044173494636, 3.03506026265434456505779216681, 3.70717818248219795825376849036, 3.87652428614993324478657608781, 4.16571129256246658456308924274, 4.56320977329341254096274747923, 5.44899571892131381918563815503, 5.46513242743369861100795806153, 5.60246499593008097503309344670, 6.21740941516161929256826489364, 6.78405464030845245417259376330, 6.84611206779404725206113332739, 7.21885384900375487185320826446, 7.67419059854830156854979116121, 7.976411683820534792778389981786, 8.225581177931808512771815524367
