| L(s) = 1 | − 3-s − 5-s − 2·9-s − 11-s − 4·13-s + 15-s + 2·17-s + 23-s − 4·25-s + 5·27-s + 7·31-s + 33-s + 3·37-s + 4·39-s + 8·41-s + 6·43-s + 2·45-s + 8·47-s − 2·51-s − 6·53-s + 55-s + 5·59-s − 12·61-s + 4·65-s + 7·67-s − 69-s + 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s + 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.25·31-s + 0.174·33-s + 0.493·37-s + 0.640·39-s + 1.24·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.134·55-s + 0.650·59-s − 1.53·61-s + 0.496·65-s + 0.855·67-s − 0.120·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.59753326202792059542705479631, −6.69785642750182140490053057916, −5.96253635220307759773371824019, −5.41592840817145656472425824771, −4.67360648409355241573689732264, −4.01118939616306336398601307633, −2.92516350065322057627024861621, −2.40668113522951484990555404309, −0.979033992832471368219893964268, 0,
0.979033992832471368219893964268, 2.40668113522951484990555404309, 2.92516350065322057627024861621, 4.01118939616306336398601307633, 4.67360648409355241573689732264, 5.41592840817145656472425824771, 5.96253635220307759773371824019, 6.69785642750182140490053057916, 7.59753326202792059542705479631
