| L(s) = 1 | − 2·5-s − 8·19-s − 25-s + 4·29-s − 16·31-s − 12·41-s − 49-s − 8·59-s − 12·61-s − 24·71-s + 8·79-s − 20·89-s + 16·95-s + 36·101-s − 28·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 32·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.83·19-s − 1/5·25-s + 0.742·29-s − 2.87·31-s − 1.87·41-s − 1/7·49-s − 1.04·59-s − 1.53·61-s − 2.84·71-s + 0.900·79-s − 2.11·89-s + 1.64·95-s + 3.58·101-s − 2.68·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4248084654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4248084654\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−10.28630686549445911423429436399, −9.181292137054726356015120717343, −9.165611770956031616213594010138, −8.643470355764417466753512147604, −8.345806656197979186835781356270, −7.72514320118517489118670067005, −7.58758503190917037834068109774, −7.08569018465165700243735778601, −6.50126589139299238862603764220, −6.35548679544745067335869699082, −5.62085455269324287305993389993, −5.30863946361300690156724032679, −4.67198866454970854600655997116, −4.12284037305519620721569663355, −4.03685911738431397278418732169, −3.18718983780593793711955095159, −2.96765216725405715849112457162, −1.79452251166668040447689198556, −1.77944218236695912773050263148, −0.26379374275432735683046553226,
0.26379374275432735683046553226, 1.77944218236695912773050263148, 1.79452251166668040447689198556, 2.96765216725405715849112457162, 3.18718983780593793711955095159, 4.03685911738431397278418732169, 4.12284037305519620721569663355, 4.67198866454970854600655997116, 5.30863946361300690156724032679, 5.62085455269324287305993389993, 6.35548679544745067335869699082, 6.50126589139299238862603764220, 7.08569018465165700243735778601, 7.58758503190917037834068109774, 7.72514320118517489118670067005, 8.345806656197979186835781356270, 8.643470355764417466753512147604, 9.165611770956031616213594010138, 9.181292137054726356015120717343, 10.28630686549445911423429436399
