L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 3·9-s − 2·10-s − 11-s − 6·13-s − 16-s + 2·17-s + 3·18-s + 8·19-s − 2·20-s + 22-s − 25-s + 6·26-s + 29-s + 4·31-s − 5·32-s − 2·34-s + 3·36-s − 2·37-s − 8·38-s + 6·40-s + 2·41-s − 4·43-s + 44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 9-s − 0.632·10-s − 0.301·11-s − 1.66·13-s − 1/4·16-s + 0.485·17-s + 0.707·18-s + 1.83·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s + 0.185·29-s + 0.718·31-s − 0.883·32-s − 0.342·34-s + 1/2·36-s − 0.328·37-s − 1.29·38-s + 0.948·40-s + 0.312·41-s − 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15631 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15631 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.62731643046730, −15.76729868714415, −15.22947862052280, −14.24088655975242, −14.04052995733013, −13.85961358274411, −12.89424707950879, −12.40321271405921, −11.74271753253412, −11.16401806163391, −10.29493779480633, −9.938935269355362, −9.429806216036498, −9.127293599759417, −8.122573787063101, −7.834735273854200, −7.187331444534128, −6.342195512496106, −5.417484486881167, −5.250318978117278, −4.543992065678343, −3.392546514676651, −2.745788818152952, −1.950839446978662, −0.9778534176548871, 0,
0.9778534176548871, 1.950839446978662, 2.745788818152952, 3.392546514676651, 4.543992065678343, 5.250318978117278, 5.417484486881167, 6.342195512496106, 7.187331444534128, 7.834735273854200, 8.122573787063101, 9.127293599759417, 9.429806216036498, 9.938935269355362, 10.29493779480633, 11.16401806163391, 11.74271753253412, 12.40321271405921, 12.89424707950879, 13.85961358274411, 14.04052995733013, 14.24088655975242, 15.22947862052280, 15.76729868714415, 16.62731643046730