| L(s) = 1 | + 3-s − 3·5-s − 7-s − 2·9-s − 5·11-s + 13-s − 3·15-s − 4·17-s + 5·19-s − 21-s + 23-s + 4·25-s − 5·27-s + 10·29-s − 5·33-s + 3·35-s + 2·37-s + 39-s + 4·41-s + 6·43-s + 6·45-s + 2·47-s + 49-s − 4·51-s − 8·53-s + 15·55-s + 5·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s − 0.774·15-s − 0.970·17-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.962·27-s + 1.85·29-s − 0.870·33-s + 0.507·35-s + 0.328·37-s + 0.160·39-s + 0.624·41-s + 0.914·43-s + 0.894·45-s + 0.291·47-s + 1/7·49-s − 0.560·51-s − 1.09·53-s + 2.02·55-s + 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.92409934191344, −15.78834196637864, −15.18427546956506, −14.61703046599984, −13.82505210027124, −13.59221673254910, −12.81168357338174, −12.34112890767361, −11.67488656568416, −11.14107895631241, −10.72907334685869, −9.949656559616578, −9.288265116477228, −8.598249718756727, −8.163816447384594, −7.715005382050030, −7.142677018276881, −6.365472539846142, −5.565249150304959, −4.895053324096266, −4.201332018672452, −3.462096441521888, −2.829054997716703, −2.415597779945456, −0.8768109171827059, 0,
0.8768109171827059, 2.415597779945456, 2.829054997716703, 3.462096441521888, 4.201332018672452, 4.895053324096266, 5.565249150304959, 6.365472539846142, 7.142677018276881, 7.715005382050030, 8.163816447384594, 8.598249718756727, 9.288265116477228, 9.949656559616578, 10.72907334685869, 11.14107895631241, 11.67488656568416, 12.34112890767361, 12.81168357338174, 13.59221673254910, 13.82505210027124, 14.61703046599984, 15.18427546956506, 15.78834196637864, 15.92409934191344
