L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 2·13-s + 15-s + 2·17-s − 19-s + 8·23-s + 25-s − 27-s − 10·29-s − 8·31-s + 4·33-s + 2·37-s − 2·39-s + 2·41-s + 8·43-s − 45-s + 8·47-s − 7·49-s − 2·51-s − 6·53-s + 4·55-s + 57-s + 4·59-s − 6·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.229·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 1.16·47-s − 49-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 0.132·57-s + 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.09140559413585, −15.53222776914553, −15.07165499333391, −14.55980100513297, −13.79736798362741, −13.07999198095416, −12.74503541008757, −12.40603209099430, −11.35337202312825, −10.99513448983788, −10.84634405172973, −9.945344499742006, −9.280332865233796, −8.801596297680178, −7.897250389776453, −7.522683010954080, −6.993850510380286, −6.129326434428164, −5.470286192696970, −5.139495990914629, −4.247129102946696, −3.585379391861314, −2.867653262185200, −1.941501017499243, −0.9447398399297686, 0,
0.9447398399297686, 1.941501017499243, 2.867653262185200, 3.585379391861314, 4.247129102946696, 5.139495990914629, 5.470286192696970, 6.129326434428164, 6.993850510380286, 7.522683010954080, 7.897250389776453, 8.801596297680178, 9.280332865233796, 9.945344499742006, 10.84634405172973, 10.99513448983788, 11.35337202312825, 12.40603209099430, 12.74503541008757, 13.07999198095416, 13.79736798362741, 14.55980100513297, 15.07165499333391, 15.53222776914553, 16.09140559413585