| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 2·15-s + 2·17-s − 4·19-s + 21-s + 6·23-s − 25-s − 27-s + 2·35-s − 2·37-s + 4·43-s − 2·45-s − 8·47-s + 49-s − 2·51-s − 4·53-s + 4·57-s − 6·59-s − 12·61-s − 63-s + 2·67-s − 6·69-s + 14·73-s + 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.338·35-s − 0.328·37-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s + 0.529·57-s − 0.781·59-s − 1.53·61-s − 0.125·63-s + 0.244·67-s − 0.722·69-s + 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7322057160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7322057160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73660624441848, −12.37993395177394, −12.18949546827599, −11.43020711218665, −11.11112294644927, −10.80434132544867, −10.18736288048764, −9.745023943540107, −9.146261617656114, −8.788208571964716, −8.077904776934086, −7.738086929287741, −7.281029641479482, −6.620611583837904, −6.367363423615247, −5.755615492782875, −5.113264076007668, −4.700064481095484, −4.188152283332685, −3.530483304697972, −3.219098895385163, −2.450008542508919, −1.714946853300250, −1.000188725556389, −0.2864529331904485,
0.2864529331904485, 1.000188725556389, 1.714946853300250, 2.450008542508919, 3.219098895385163, 3.530483304697972, 4.188152283332685, 4.700064481095484, 5.113264076007668, 5.755615492782875, 6.367363423615247, 6.620611583837904, 7.281029641479482, 7.738086929287741, 8.077904776934086, 8.788208571964716, 9.146261617656114, 9.745023943540107, 10.18736288048764, 10.80434132544867, 11.11112294644927, 11.43020711218665, 12.18949546827599, 12.37993395177394, 12.73660624441848
