L(s) = 1 | + 2-s + 3·3-s − 4-s − 5-s + 3·6-s − 3·8-s + 6·9-s − 10-s − 3·12-s + 4·13-s − 3·15-s − 16-s + 6·18-s + 4·19-s + 20-s − 8·23-s − 9·24-s + 25-s + 4·26-s + 9·27-s − 6·29-s − 3·30-s + 2·31-s + 5·32-s − 6·36-s − 8·37-s + 4·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.447·5-s + 1.22·6-s − 1.06·8-s + 2·9-s − 0.316·10-s − 0.866·12-s + 1.10·13-s − 0.774·15-s − 1/4·16-s + 1.41·18-s + 0.917·19-s + 0.223·20-s − 1.66·23-s − 1.83·24-s + 1/5·25-s + 0.784·26-s + 1.73·27-s − 1.11·29-s − 0.547·30-s + 0.359·31-s + 0.883·32-s − 36-s − 1.31·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−15.34230598622451, −14.73429737991480, −14.27260017959469, −13.77560683790079, −13.47339340592787, −13.14304469021291, −12.27532025996541, −11.98498506234875, −11.29821585991569, −10.29914706830640, −10.01703322607418, −9.208427204871347, −8.872724478363872, −8.399439649549005, −7.855954557246424, −7.369678678465814, −6.564147255146451, −5.834184041011941, −5.213630653914079, −4.266545684921945, −4.011351935412016, −3.299051842285343, −3.088748942569854, −2.024215821089865, −1.344780598710011, 0,
1.344780598710011, 2.024215821089865, 3.088748942569854, 3.299051842285343, 4.011351935412016, 4.266545684921945, 5.213630653914079, 5.834184041011941, 6.564147255146451, 7.369678678465814, 7.855954557246424, 8.399439649549005, 8.872724478363872, 9.208427204871347, 10.01703322607418, 10.29914706830640, 11.29821585991569, 11.98498506234875, 12.27532025996541, 13.14304469021291, 13.47339340592787, 13.77560683790079, 14.27260017959469, 14.73429737991480, 15.34230598622451