| L(s) = 1 | − 4-s + 2·11-s + 16-s + 2·19-s + 12·29-s − 20·31-s − 18·41-s − 2·44-s + 13·49-s − 6·59-s − 8·61-s − 64-s − 6·71-s − 2·76-s − 22·79-s − 18·101-s + 32·109-s − 12·116-s + 3·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.603·11-s + 1/4·16-s + 0.458·19-s + 2.22·29-s − 3.59·31-s − 2.81·41-s − 0.301·44-s + 13/7·49-s − 0.781·59-s − 1.02·61-s − 1/8·64-s − 0.712·71-s − 0.229·76-s − 2.47·79-s − 1.79·101-s + 3.06·109-s − 1.11·116-s + 3/11·121-s + 1.79·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098288647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098288647\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−8.573252620512681023737960310641, −8.145555130971863829020683906994, −7.70992627230976277291876987976, −7.15099226045158239509323945878, −7.10794908718810505118931117719, −6.83276997121571623484277376604, −6.13711983539545958874666755361, −5.87761279426441816734915076136, −5.55192212151340425180336406129, −5.14392204964496373520259899107, −4.62766455404119462040891807410, −4.52217431374977024233741130919, −3.82668162435503891309921667805, −3.59981930051105400914909765394, −3.15766780160468425863067446439, −2.76755371694855411132321440023, −2.02281779539505151343997399276, −1.60505730395072020941294351075, −1.16661260109048658520750973745, −0.30081103510883507960019505885,
0.30081103510883507960019505885, 1.16661260109048658520750973745, 1.60505730395072020941294351075, 2.02281779539505151343997399276, 2.76755371694855411132321440023, 3.15766780160468425863067446439, 3.59981930051105400914909765394, 3.82668162435503891309921667805, 4.52217431374977024233741130919, 4.62766455404119462040891807410, 5.14392204964496373520259899107, 5.55192212151340425180336406129, 5.87761279426441816734915076136, 6.13711983539545958874666755361, 6.83276997121571623484277376604, 7.10794908718810505118931117719, 7.15099226045158239509323945878, 7.70992627230976277291876987976, 8.145555130971863829020683906994, 8.573252620512681023737960310641
