| L(s) = 1 | + 2·3-s + 3·9-s − 12·23-s + 2·25-s + 4·27-s + 12·29-s − 2·43-s − 11·49-s + 24·53-s + 2·61-s − 24·69-s + 4·75-s + 22·79-s + 5·81-s + 24·87-s + 36·101-s − 2·103-s + 12·107-s + 12·113-s − 10·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 22·147-s + 149-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.50·23-s + 2/5·25-s + 0.769·27-s + 2.22·29-s − 0.304·43-s − 1.57·49-s + 3.29·53-s + 0.256·61-s − 2.88·69-s + 0.461·75-s + 2.47·79-s + 5/9·81-s + 2.57·87-s + 3.58·101-s − 0.197·103-s + 1.16·107-s + 1.12·113-s − 0.909·121-s + 0.0887·127-s − 0.352·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.81·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.524496206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.524496206\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.914742258493993233136385732467, −7.78573258930618977651538713247, −7.48636729143301806935570681592, −6.91902854097483798828469619614, −6.67378145344098308048578793339, −6.35644948737587482518629829947, −6.01124298476723272419499876157, −5.61078665261345951603692006958, −5.08143603512056640526757292127, −4.79907325999365744746282630217, −4.29965369160807394585883922723, −4.15786349006164698609965746421, −3.48934529834465575486508023284, −3.45678724041477860448606587734, −2.85567185138923397641951261937, −2.43042029089779197788124239944, −2.00579167201248311567823960584, −1.82581884991847559444437499937, −0.931919197640142301796455455128, −0.58609583419050332989630214613,
0.58609583419050332989630214613, 0.931919197640142301796455455128, 1.82581884991847559444437499937, 2.00579167201248311567823960584, 2.43042029089779197788124239944, 2.85567185138923397641951261937, 3.45678724041477860448606587734, 3.48934529834465575486508023284, 4.15786349006164698609965746421, 4.29965369160807394585883922723, 4.79907325999365744746282630217, 5.08143603512056640526757292127, 5.61078665261345951603692006958, 6.01124298476723272419499876157, 6.35644948737587482518629829947, 6.67378145344098308048578793339, 6.91902854097483798828469619614, 7.48636729143301806935570681592, 7.78573258930618977651538713247, 7.914742258493993233136385732467
