| L(s) = 1 | + 4·17-s − 6·25-s + 12·29-s − 8·43-s + 10·49-s + 20·53-s − 28·61-s − 32·79-s + 20·101-s + 16·103-s − 24·107-s − 12·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 0.970·17-s − 6/5·25-s + 2.22·29-s − 1.21·43-s + 10/7·49-s + 2.74·53-s − 3.58·61-s − 3.60·79-s + 1.99·101-s + 1.57·103-s − 2.32·107-s − 1.12·113-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37015056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112371867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112371867\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 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
−8.230495974696161556968800712705, −7.86883675925743340619539735582, −7.51827146527185353221507447546, −7.28506860434039394541854255566, −6.81339136632001866902391491031, −6.53583243610749820207648519736, −5.91252921501602627283122945883, −5.90829608192372488231102311456, −5.47337383370907290320225177554, −4.95512404778320835572716675827, −4.67939472256360265681134442419, −4.12866678317033503512860650176, −3.98394121283418240112609938330, −3.38585061432719844416773003867, −2.88918718127817437261926638706, −2.72874240225514773876500529169, −2.10265541506396295092867596618, −1.47599743442011029036513124347, −1.14490195134055273114447289803, −0.39530859508133971814905777578,
0.39530859508133971814905777578, 1.14490195134055273114447289803, 1.47599743442011029036513124347, 2.10265541506396295092867596618, 2.72874240225514773876500529169, 2.88918718127817437261926638706, 3.38585061432719844416773003867, 3.98394121283418240112609938330, 4.12866678317033503512860650176, 4.67939472256360265681134442419, 4.95512404778320835572716675827, 5.47337383370907290320225177554, 5.90829608192372488231102311456, 5.91252921501602627283122945883, 6.53583243610749820207648519736, 6.81339136632001866902391491031, 7.28506860434039394541854255566, 7.51827146527185353221507447546, 7.86883675925743340619539735582, 8.230495974696161556968800712705
