| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s − 8·19-s − 12·29-s − 2·31-s + 36-s + 20·41-s + 8·44-s + 14·49-s + 24·59-s − 4·61-s − 64-s + 8·76-s + 81-s + 28·89-s + 8·99-s + 12·101-s + 36·109-s + 12·116-s + 26·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s − 1.83·19-s − 2.22·29-s − 0.359·31-s + 1/6·36-s + 3.12·41-s + 1.20·44-s + 2·49-s + 3.12·59-s − 0.512·61-s − 1/8·64-s + 0.917·76-s + 1/9·81-s + 2.96·89-s + 0.804·99-s + 1.19·101-s + 3.44·109-s + 1.11·116-s + 2.36·121-s + 0.179·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21622500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238427401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238427401\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.525647481673992696530012265165, −8.131172091433443907611229110211, −7.64225774108516106770381217401, −7.52881854441892907560908322998, −7.28985638622406679814529895354, −6.73066408545719865337403776170, −6.03265036783062671656038008801, −5.84179288751046260251331246618, −5.73308360394574246755242445904, −5.05503011365676459693430618962, −4.97036767889182386816932168076, −4.36700328846565953683990333043, −3.93407613333581363368019575718, −3.72116992769178786307264091475, −3.06084798008186798510762988705, −2.51835469249194975979516482251, −2.10672157202498530092111965838, −2.10071159374473486521558737460, −0.75200811348567704805352024882, −0.42491837607475486422221526359,
0.42491837607475486422221526359, 0.75200811348567704805352024882, 2.10071159374473486521558737460, 2.10672157202498530092111965838, 2.51835469249194975979516482251, 3.06084798008186798510762988705, 3.72116992769178786307264091475, 3.93407613333581363368019575718, 4.36700328846565953683990333043, 4.97036767889182386816932168076, 5.05503011365676459693430618962, 5.73308360394574246755242445904, 5.84179288751046260251331246618, 6.03265036783062671656038008801, 6.73066408545719865337403776170, 7.28985638622406679814529895354, 7.52881854441892907560908322998, 7.64225774108516106770381217401, 8.131172091433443907611229110211, 8.525647481673992696530012265165
