| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s + 2·13-s + 15-s − 6·17-s + 4·19-s − 21-s − 23-s + 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s − 35-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s + 49-s + 6·51-s − 6·53-s + 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.34127985852297, −14.64256148441794, −13.89203365288239, −13.40897804137746, −13.17187581970933, −12.32211404726454, −11.95466628092538, −11.37249184607016, −10.88620133467872, −10.57301441989243, −9.858887982951051, −9.306238002841320, −8.531141608406281, −8.106737552861007, −7.625764606768142, −6.929189723898253, −6.402534501222229, −5.759929856529331, −5.158143769993203, −4.517032928521230, −4.206889844912944, −3.136090618165696, −2.649925622574442, −1.735511341512926, −0.8623442710072221, 0,
0.8623442710072221, 1.735511341512926, 2.649925622574442, 3.136090618165696, 4.206889844912944, 4.517032928521230, 5.158143769993203, 5.759929856529331, 6.402534501222229, 6.929189723898253, 7.625764606768142, 8.106737552861007, 8.531141608406281, 9.306238002841320, 9.858887982951051, 10.57301441989243, 10.88620133467872, 11.37249184607016, 11.95466628092538, 12.32211404726454, 13.17187581970933, 13.40897804137746, 13.89203365288239, 14.64256148441794, 15.34127985852297
