| L(s) = 1 | + 4·5-s − 2·7-s + 4·11-s + 13-s − 2·17-s − 2·19-s + 11·25-s + 6·29-s − 10·31-s − 8·35-s + 10·37-s − 8·41-s + 4·43-s + 4·47-s − 3·49-s + 10·53-s + 16·55-s + 8·59-s − 14·61-s + 4·65-s + 2·67-s − 16·71-s − 10·73-s − 8·77-s − 16·79-s − 8·85-s + 4·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.458·19-s + 11/5·25-s + 1.11·29-s − 1.79·31-s − 1.35·35-s + 1.64·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s + 1.37·53-s + 2.15·55-s + 1.04·59-s − 1.79·61-s + 0.496·65-s + 0.244·67-s − 1.89·71-s − 1.17·73-s − 0.911·77-s − 1.80·79-s − 0.867·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.826951888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826951888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−10.81609252174330783275568401880, −10.02637231667649988922232878626, −9.272613788586316089970782456314, −8.740006195634140431482743685580, −7.00743292844584939053022110799, −6.30352193131009441178367980896, −5.63885615575172245866908586348, −4.22765636287388069600769197082, −2.78430186696517884115720808997, −1.53061467325951463049581429920,
1.53061467325951463049581429920, 2.78430186696517884115720808997, 4.22765636287388069600769197082, 5.63885615575172245866908586348, 6.30352193131009441178367980896, 7.00743292844584939053022110799, 8.740006195634140431482743685580, 9.272613788586316089970782456314, 10.02637231667649988922232878626, 10.81609252174330783275568401880
