| L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s + 7-s − 3·8-s + 9-s + 2·10-s − 11-s + 12-s + 4·13-s + 14-s − 2·15-s − 16-s + 17-s + 18-s − 19-s − 2·20-s − 21-s − 22-s − 9·23-s + 3·24-s − 25-s + 4·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s − 1.87·23-s + 0.612·24-s − 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17661 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.96201099653654, −15.61190956826785, −14.87513323355417, −14.25964622651737, −13.83627490845937, −13.39653125822063, −12.98545635960837, −12.20152772338301, −11.87121076499978, −11.18945020187528, −10.43022990182343, −10.04109776030711, −9.454153123870662, −8.687274021245606, −8.255411682097387, −7.488221281596912, −6.501054228513155, −6.040840192649684, −5.590608873457005, −5.136372204858471, −4.080514565400712, −3.998936056797060, −2.856938696638001, −2.015038760540274, −1.163060100240316, 0,
1.163060100240316, 2.015038760540274, 2.856938696638001, 3.998936056797060, 4.080514565400712, 5.136372204858471, 5.590608873457005, 6.040840192649684, 6.501054228513155, 7.488221281596912, 8.255411682097387, 8.687274021245606, 9.454153123870662, 10.04109776030711, 10.43022990182343, 11.18945020187528, 11.87121076499978, 12.20152772338301, 12.98545635960837, 13.39653125822063, 13.83627490845937, 14.25964622651737, 14.87513323355417, 15.61190956826785, 15.96201099653654
