| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 11-s − 12-s − 4·13-s − 14-s + 16-s − 18-s − 4·19-s − 21-s − 22-s − 6·23-s + 24-s + 4·26-s − 27-s + 28-s − 2·29-s + 8·31-s − 32-s − 33-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.174·33-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−16.83578122500735, −16.24086613249227, −15.75569257669040, −15.06514978604266, −14.55341945008722, −14.01087478218705, −13.19066401010071, −12.42089420371896, −12.09958386096704, −11.50987151141501, −10.86187442486607, −10.34814995143019, −9.729847960106115, −9.275885218387727, −8.321229009839671, −7.994178807282505, −7.175605166340783, −6.656755122095941, −5.946514340770904, −5.309560778067839, −4.439380932578861, −3.908715528219251, −2.610613331313716, −2.068283892992480, −1.007612735100423, 0,
1.007612735100423, 2.068283892992480, 2.610613331313716, 3.908715528219251, 4.439380932578861, 5.309560778067839, 5.946514340770904, 6.656755122095941, 7.175605166340783, 7.994178807282505, 8.321229009839671, 9.275885218387727, 9.729847960106115, 10.34814995143019, 10.86187442486607, 11.50987151141501, 12.09958386096704, 12.42089420371896, 13.19066401010071, 14.01087478218705, 14.55341945008722, 15.06514978604266, 15.75569257669040, 16.24086613249227, 16.83578122500735
