| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 5·7-s − 8-s + 9-s + 3·10-s − 5·11-s − 12-s − 6·13-s + 5·14-s + 3·15-s + 16-s − 4·17-s − 18-s − 6·19-s − 3·20-s + 5·21-s + 5·22-s − 9·23-s + 24-s + 4·25-s + 6·26-s − 27-s − 5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.50·11-s − 0.288·12-s − 1.66·13-s + 1.33·14-s + 0.774·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.670·20-s + 1.09·21-s + 1.06·22-s − 1.87·23-s + 0.204·24-s + 4/5·25-s + 1.17·26-s − 0.192·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 4091 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.31528206139947, −15.67280215818628, −15.27765806666043, −15.13764688693100, −14.04077813169368, −13.21607344797135, −12.70678039238815, −12.43186175997173, −11.99957553692154, −11.12802351472510, −10.79347345852382, −10.08832008075510, −9.824393236399667, −9.136281439573843, −8.404230195222007, −7.775440278968898, −7.313595578356126, −6.865557401328231, −6.181236874744949, −5.581468262683396, −4.737631748021957, −4.009415906944245, −3.461691784247171, −2.526933557444731, −2.093900260196079, 0, 0, 0,
2.093900260196079, 2.526933557444731, 3.461691784247171, 4.009415906944245, 4.737631748021957, 5.581468262683396, 6.181236874744949, 6.865557401328231, 7.313595578356126, 7.775440278968898, 8.404230195222007, 9.136281439573843, 9.824393236399667, 10.08832008075510, 10.79347345852382, 11.12802351472510, 11.99957553692154, 12.43186175997173, 12.70678039238815, 13.21607344797135, 14.04077813169368, 15.13764688693100, 15.27765806666043, 15.67280215818628, 16.31528206139947
