| L(s) = 1 | − 2·4-s − 5-s + 7-s − 11-s + 4·16-s − 2·17-s + 19-s + 2·20-s + 5·23-s − 4·25-s − 2·28-s + 5·29-s + 9·31-s − 35-s − 10·37-s − 2·41-s − 13·43-s + 2·44-s − 7·47-s + 49-s + 53-s + 55-s + 8·59-s − 2·61-s − 8·64-s − 2·67-s + 4·68-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s − 0.301·11-s + 16-s − 0.485·17-s + 0.229·19-s + 0.447·20-s + 1.04·23-s − 4/5·25-s − 0.377·28-s + 0.928·29-s + 1.61·31-s − 0.169·35-s − 1.64·37-s − 0.312·41-s − 1.98·43-s + 0.301·44-s − 1.02·47-s + 1/7·49-s + 0.137·53-s + 0.134·55-s + 1.04·59-s − 0.256·61-s − 64-s − 0.244·67-s + 0.485·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.68635941502214, −13.40062614972723, −13.07743455326176, −12.24236226656069, −11.97135507094261, −11.55129692553381, −10.85625959148859, −10.36789246345256, −9.931776129849536, −9.470652707128217, −8.773076258057578, −8.354829298114155, −8.158051867655013, −7.488777327036579, −6.766900344088686, −6.487066076579021, −5.524829215202377, −5.157249094244661, −4.643248311515523, −4.279381892083020, −3.367855993950503, −3.218190399488782, −2.235731189739084, −1.479227474290630, −0.7421515471861383, 0,
0.7421515471861383, 1.479227474290630, 2.235731189739084, 3.218190399488782, 3.367855993950503, 4.279381892083020, 4.643248311515523, 5.157249094244661, 5.524829215202377, 6.487066076579021, 6.766900344088686, 7.488777327036579, 8.158051867655013, 8.354829298114155, 8.773076258057578, 9.470652707128217, 9.931776129849536, 10.36789246345256, 10.85625959148859, 11.55129692553381, 11.97135507094261, 12.24236226656069, 13.07743455326176, 13.40062614972723, 13.68635941502214
