| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 12-s − 4·13-s − 2·14-s + 16-s − 6·17-s − 18-s + 4·19-s − 2·21-s − 6·23-s + 24-s + 4·26-s − 27-s + 2·28-s − 6·29-s + 8·31-s − 32-s + 6·34-s + 36-s + 10·37-s − 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.436·21-s − 1.25·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8433000708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8433000708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.83921548228591, −15.38747587225585, −14.84768867925781, −14.17970097337979, −13.64548936037945, −12.95017606133921, −12.21815841715427, −11.84840531346874, −11.21967643697253, −10.93531579807689, −10.09645847043574, −9.650612194534090, −9.172686227873408, −8.252304364355296, −7.905591873485777, −7.214506467005306, −6.722345246158730, −5.900799202432833, −5.416075755736886, −4.478705838911386, −4.218694145481117, −2.889675108848370, −2.239767330633625, −1.483764006939233, −0.4509071286736605,
0.4509071286736605, 1.483764006939233, 2.239767330633625, 2.889675108848370, 4.218694145481117, 4.478705838911386, 5.416075755736886, 5.900799202432833, 6.722345246158730, 7.214506467005306, 7.905591873485777, 8.252304364355296, 9.172686227873408, 9.650612194534090, 10.09645847043574, 10.93531579807689, 11.21967643697253, 11.84840531346874, 12.21815841715427, 12.95017606133921, 13.64548936037945, 14.17970097337979, 14.84768867925781, 15.38747587225585, 15.83921548228591
