| L(s) = 1 | − 3·3-s − 2·5-s + 2·7-s + 6·9-s − 5·11-s + 4·13-s + 6·15-s + 2·17-s − 10·19-s − 6·21-s + 4·23-s + 5·25-s − 9·27-s − 6·29-s + 15·33-s − 4·35-s + 20·37-s − 12·39-s + 3·41-s + 9·43-s − 12·45-s − 8·47-s + 7·49-s − 6·51-s + 24·53-s + 10·55-s + 30·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s + 0.755·7-s + 2·9-s − 1.50·11-s + 1.10·13-s + 1.54·15-s + 0.485·17-s − 2.29·19-s − 1.30·21-s + 0.834·23-s + 25-s − 1.73·27-s − 1.11·29-s + 2.61·33-s − 0.676·35-s + 3.28·37-s − 1.92·39-s + 0.468·41-s + 1.37·43-s − 1.78·45-s − 1.16·47-s + 49-s − 0.840·51-s + 3.29·53-s + 1.34·55-s + 3.97·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8700805792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8700805792\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.05269705339901852894008974909, −9.968646117772716200464282202599, −8.976563748980780377745863618160, −8.825993875729138970748968380421, −8.197951320930899893405917015520, −7.983430496040446578006628728183, −7.36789985313703095915963689758, −7.23224505009236497278634632527, −6.62414820290657301474689088788, −5.97066375407891166138338075674, −5.82184596543188458367752445307, −5.50866357116987469866208978767, −4.61050929935851078248876764669, −4.60471357704948884024862892975, −4.14233055175990686205372118648, −3.55006983724502924308295996107, −2.62920653369505239962300720912, −2.14979699937161635144707756177, −1.06692554332334423205375008396, −0.56048268052221173417437057464,
0.56048268052221173417437057464, 1.06692554332334423205375008396, 2.14979699937161635144707756177, 2.62920653369505239962300720912, 3.55006983724502924308295996107, 4.14233055175990686205372118648, 4.60471357704948884024862892975, 4.61050929935851078248876764669, 5.50866357116987469866208978767, 5.82184596543188458367752445307, 5.97066375407891166138338075674, 6.62414820290657301474689088788, 7.23224505009236497278634632527, 7.36789985313703095915963689758, 7.983430496040446578006628728183, 8.197951320930899893405917015520, 8.825993875729138970748968380421, 8.976563748980780377745863618160, 9.968646117772716200464282202599, 10.05269705339901852894008974909
