| L(s) = 1 | − 4-s + 2·5-s + 10·11-s + 16-s + 10·19-s − 2·20-s − 25-s − 4·29-s + 2·31-s − 12·41-s − 10·44-s + 13·49-s + 20·55-s + 20·59-s − 28·61-s − 64-s + 18·71-s − 10·76-s − 10·79-s + 2·80-s + 6·89-s + 20·95-s + 100-s + 14·101-s + 4·109-s + 4·116-s + 53·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 3.01·11-s + 1/4·16-s + 2.29·19-s − 0.447·20-s − 1/5·25-s − 0.742·29-s + 0.359·31-s − 1.87·41-s − 1.50·44-s + 13/7·49-s + 2.69·55-s + 2.60·59-s − 3.58·61-s − 1/8·64-s + 2.13·71-s − 1.14·76-s − 1.12·79-s + 0.223·80-s + 0.635·89-s + 2.05·95-s + 1/10·100-s + 1.39·101-s + 0.383·109-s + 0.371·116-s + 4.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.191904832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.191904832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−9.002723344610718315109962323414, −8.892859740990070057605341090576, −8.393515000752099888774058015152, −7.81525480410426577040642305034, −7.38757921851535832777826796460, −7.11977385300912427656373129545, −6.61190049229403080180614047753, −6.39019997546870247413441295472, −5.89317436261654463177875560063, −5.59504673304903008351887190320, −5.13959725664855712548054563136, −4.77590708507078586646068419023, −4.02291739456649782699085467525, −3.95894335343993500232205444459, −3.38627285382185020667461616799, −3.08728682392209499772947350703, −2.20270623538410678589195443290, −1.66827752237531685909395942898, −1.25650241880376665435660990080, −0.75867701375121566635650294198,
0.75867701375121566635650294198, 1.25650241880376665435660990080, 1.66827752237531685909395942898, 2.20270623538410678589195443290, 3.08728682392209499772947350703, 3.38627285382185020667461616799, 3.95894335343993500232205444459, 4.02291739456649782699085467525, 4.77590708507078586646068419023, 5.13959725664855712548054563136, 5.59504673304903008351887190320, 5.89317436261654463177875560063, 6.39019997546870247413441295472, 6.61190049229403080180614047753, 7.11977385300912427656373129545, 7.38757921851535832777826796460, 7.81525480410426577040642305034, 8.393515000752099888774058015152, 8.892859740990070057605341090576, 9.002723344610718315109962323414
