| L(s) = 1 | − 2·5-s − 4·7-s − 2·11-s − 8·19-s + 3·25-s + 4·29-s − 4·31-s + 8·35-s + 4·37-s + 4·41-s − 4·43-s + 8·47-s + 4·53-s + 4·55-s + 12·59-s + 12·61-s + 4·71-s + 8·77-s − 16·79-s + 12·83-s + 4·89-s + 16·95-s + 12·97-s + 20·101-s + 12·107-s − 4·109-s + 4·113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.603·11-s − 1.83·19-s + 3/5·25-s + 0.742·29-s − 0.718·31-s + 1.35·35-s + 0.657·37-s + 0.624·41-s − 0.609·43-s + 1.16·47-s + 0.549·53-s + 0.539·55-s + 1.56·59-s + 1.53·61-s + 0.474·71-s + 0.911·77-s − 1.80·79-s + 1.31·83-s + 0.423·89-s + 1.64·95-s + 1.21·97-s + 1.99·101-s + 1.16·107-s − 0.383·109-s + 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.217561855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.217561855\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 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
−8.606088853871438762727263899983, −8.315436329354925363242536196633, −7.897752765158683490692135554973, −7.59123582022504381012651933355, −7.05007267462379839584269795809, −6.89069243100507081497991434046, −6.42726327278666507722018433274, −6.20028629326285780889600834027, −5.72418659637057702562248655727, −5.32239700098825477563651214801, −4.82908027777640229947502383782, −4.31200300994431665749310696553, −4.10045471059502583758509609818, −3.62941404009603366868147852602, −3.19897247149752880576649682272, −2.84705963764479627571533574096, −2.23213888462537797815943033338, −1.95474066511156490144334047948, −0.72567876661307557380803438624, −0.47043807554754514671534769534,
0.47043807554754514671534769534, 0.72567876661307557380803438624, 1.95474066511156490144334047948, 2.23213888462537797815943033338, 2.84705963764479627571533574096, 3.19897247149752880576649682272, 3.62941404009603366868147852602, 4.10045471059502583758509609818, 4.31200300994431665749310696553, 4.82908027777640229947502383782, 5.32239700098825477563651214801, 5.72418659637057702562248655727, 6.20028629326285780889600834027, 6.42726327278666507722018433274, 6.89069243100507081497991434046, 7.05007267462379839584269795809, 7.59123582022504381012651933355, 7.897752765158683490692135554973, 8.315436329354925363242536196633, 8.606088853871438762727263899983
