| L(s) = 1 | − 2·2-s + 3·4-s − 5-s + 2·7-s − 4·8-s + 2·10-s + 5·11-s + 6·13-s − 4·14-s + 5·16-s + 2·19-s − 3·20-s − 10·22-s + 2·23-s − 6·25-s − 12·26-s + 6·28-s + 9·29-s − 6·32-s − 2·35-s + 37-s − 4·38-s + 4·40-s − 41-s − 9·43-s + 15·44-s − 4·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s − 1.41·8-s + 0.632·10-s + 1.50·11-s + 1.66·13-s − 1.06·14-s + 5/4·16-s + 0.458·19-s − 0.670·20-s − 2.13·22-s + 0.417·23-s − 6/5·25-s − 2.35·26-s + 1.13·28-s + 1.67·29-s − 1.06·32-s − 0.338·35-s + 0.164·37-s − 0.648·38-s + 0.632·40-s − 0.156·41-s − 1.37·43-s + 2.26·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.929712764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.929712764\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 25 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 49 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 79 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 157 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 99 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 169 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19 T + 219 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 259 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23 T + 323 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
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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
−8.974173038857062072842449375046, −8.708768568113937806263205065740, −8.398576021294036797654464323804, −8.278492448305657074646989458852, −7.59892525676571574826897773990, −7.51378807125120475668669471345, −6.90444656238237957179014017141, −6.41419493288544355415798987996, −6.30542017675326017872311343331, −5.97801294122048653218172863006, −5.04942005579579044149148381052, −5.02830820173614971157011356609, −4.15361547119421797286383868612, −3.82922632677736012804267901955, −3.39590597456528441746154238291, −2.97692665940707432912947827403, −1.92281314483725312982576680740, −1.87213112785340642997889297475, −0.983447341182025989187843292750, −0.78852601037314634742331791166,
0.78852601037314634742331791166, 0.983447341182025989187843292750, 1.87213112785340642997889297475, 1.92281314483725312982576680740, 2.97692665940707432912947827403, 3.39590597456528441746154238291, 3.82922632677736012804267901955, 4.15361547119421797286383868612, 5.02830820173614971157011356609, 5.04942005579579044149148381052, 5.97801294122048653218172863006, 6.30542017675326017872311343331, 6.41419493288544355415798987996, 6.90444656238237957179014017141, 7.51378807125120475668669471345, 7.59892525676571574826897773990, 8.278492448305657074646989458852, 8.398576021294036797654464323804, 8.708768568113937806263205065740, 8.974173038857062072842449375046
