| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 4·11-s − 2·13-s − 15-s − 6·17-s + 19-s − 4·21-s − 4·23-s + 25-s + 27-s − 6·29-s − 8·31-s − 4·33-s + 4·35-s − 10·37-s − 2·39-s + 2·41-s − 45-s + 4·47-s + 9·49-s − 6·51-s − 2·53-s + 4·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.229·19-s − 0.872·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.676·35-s − 1.64·37-s − 0.320·39-s + 0.312·41-s − 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.05420111553058, −15.86165928994557, −15.32254999127757, −14.85723963773269, −14.00497408742435, −13.55717399832565, −12.97464999531185, −12.65039993026652, −12.13828260384628, −11.22179167694752, −10.72520392003062, −10.10988585414180, −9.596432452029700, −8.984496747249288, −8.564507701307244, −7.606389794438734, −7.327505832944660, −6.706535405152223, −5.925256226309864, −5.301229639885858, −4.442305452698259, −3.768097333304986, −3.170016058430164, −2.510980386056534, −1.818188674800310, 0, 0,
1.818188674800310, 2.510980386056534, 3.170016058430164, 3.768097333304986, 4.442305452698259, 5.301229639885858, 5.925256226309864, 6.706535405152223, 7.327505832944660, 7.606389794438734, 8.564507701307244, 8.984496747249288, 9.596432452029700, 10.10988585414180, 10.72520392003062, 11.22179167694752, 12.13828260384628, 12.65039993026652, 12.97464999531185, 13.55717399832565, 14.00497408742435, 14.85723963773269, 15.32254999127757, 15.86165928994557, 16.05420111553058
