| L(s) = 1 | + 5-s + 3·7-s + 5·11-s − 4·13-s − 5·17-s + 8·19-s + 6·23-s + 25-s + 4·29-s − 6·31-s + 3·35-s + 4·37-s − 10·41-s − 7·43-s − 8·47-s + 2·49-s − 53-s + 5·55-s − 4·59-s + 5·61-s − 4·65-s + 3·67-s − 13·71-s − 2·73-s + 15·77-s − 12·83-s − 5·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 1.50·11-s − 1.10·13-s − 1.21·17-s + 1.83·19-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 1.07·31-s + 0.507·35-s + 0.657·37-s − 1.56·41-s − 1.06·43-s − 1.16·47-s + 2/7·49-s − 0.137·53-s + 0.674·55-s − 0.520·59-s + 0.640·61-s − 0.496·65-s + 0.366·67-s − 1.54·71-s − 0.234·73-s + 1.70·77-s − 1.31·83-s − 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.24355972628102, −13.94811195923594, −13.28416328689958, −12.89347033095390, −12.09455848866091, −11.72148492192958, −11.36759445409305, −10.98155775892504, −10.12388127310894, −9.714430969500914, −9.270816542350004, −8.710251744239861, −8.327333003818525, −7.502391666182951, −7.043673926505854, −6.733091079335472, −5.984381312331011, −5.228598754700745, −4.874281347329583, −4.487967751776086, −3.599507684757074, −3.025971654680535, −2.285727754186348, −1.445576757171662, −1.281152342985897, 0,
1.281152342985897, 1.445576757171662, 2.285727754186348, 3.025971654680535, 3.599507684757074, 4.487967751776086, 4.874281347329583, 5.228598754700745, 5.984381312331011, 6.733091079335472, 7.043673926505854, 7.502391666182951, 8.327333003818525, 8.710251744239861, 9.270816542350004, 9.714430969500914, 10.12388127310894, 10.98155775892504, 11.36759445409305, 11.72148492192958, 12.09455848866091, 12.89347033095390, 13.28416328689958, 13.94811195923594, 14.24355972628102
