| L(s) = 1 | − 2·2-s + 4·4-s + 6·5-s − 7·7-s − 16·8-s − 12·10-s + 10·11-s + 14·14-s + 32·16-s + 12·17-s + 57·19-s + 24·20-s − 20·22-s + 40·23-s − 25-s − 28·28-s − 32·29-s + 9·31-s − 64·32-s − 24·34-s − 42·35-s − 5·37-s − 114·38-s − 96·40-s − 38·43-s + 40·44-s − 80·46-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 6/5·5-s − 7-s − 2·8-s − 6/5·10-s + 0.909·11-s + 14-s + 2·16-s + 0.705·17-s + 3·19-s + 6/5·20-s − 0.909·22-s + 1.73·23-s − 0.0399·25-s − 28-s − 1.10·29-s + 9/31·31-s − 2·32-s − 0.705·34-s − 6/5·35-s − 0.135·37-s − 3·38-s − 2.39·40-s − 0.883·43-s + 0.909·44-s − 1.73·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9866900777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9866900777\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 6 T + 37 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T - 21 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 337 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 40 T + 1071 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 988 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 1344 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 19 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T + 4909 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 32 T - 1785 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 72 T + 5209 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 36 T + 4153 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 59 T - 1008 T^{2} + 59 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 33 T + 5692 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 47 T - 4032 T^{2} + 47 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13190 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 204 T + 21793 T^{2} + 204 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16466 T^{2} + p^{4} T^{4} \) |
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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
−15.20682151561462205079073779621, −14.36388323686347471253844365465, −14.04155575772000697237772865330, −13.29864896912335229859344788035, −12.74998536901454311563234903744, −11.90969665181931988904984600347, −11.79601124933233342885795450907, −10.97634116785294090208064560864, −10.08696605483776754907859360005, −9.655342920557803296324139533624, −9.187758856964400880568009922931, −9.134148080496635979086126326722, −7.87505601997023072789746656633, −7.10202213102005220828370999701, −6.62489361510316927389053354926, −5.62909091070315445450855641686, −5.57287677313616133854472190198, −3.40400231964604647511321347250, −2.91051489452654255124263516881, −1.25149660201208419863402630597,
1.25149660201208419863402630597, 2.91051489452654255124263516881, 3.40400231964604647511321347250, 5.57287677313616133854472190198, 5.62909091070315445450855641686, 6.62489361510316927389053354926, 7.10202213102005220828370999701, 7.87505601997023072789746656633, 9.134148080496635979086126326722, 9.187758856964400880568009922931, 9.655342920557803296324139533624, 10.08696605483776754907859360005, 10.97634116785294090208064560864, 11.79601124933233342885795450907, 11.90969665181931988904984600347, 12.74998536901454311563234903744, 13.29864896912335229859344788035, 14.04155575772000697237772865330, 14.36388323686347471253844365465, 15.20682151561462205079073779621
