| L(s) = 1 | − 5-s − 13-s + 6·17-s + 4·19-s + 25-s − 2·29-s + 2·37-s + 2·41-s − 4·43-s + 4·47-s − 7·49-s − 10·53-s − 8·59-s + 2·61-s + 65-s − 4·67-s − 12·71-s − 6·73-s + 16·83-s − 6·85-s + 10·89-s − 4·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.277·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.583·47-s − 49-s − 1.37·53-s − 1.04·59-s + 0.256·61-s + 0.124·65-s − 0.488·67-s − 1.42·71-s − 0.702·73-s + 1.75·83-s − 0.650·85-s + 1.05·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.96518277248433, −14.66296920691594, −14.18403814363642, −13.55681294354865, −13.04594094125066, −12.39437039415526, −11.97782273269428, −11.60265227411908, −10.87995629751232, −10.43310899446187, −9.681658412610008, −9.451565062330952, −8.696336172288354, −7.975375580761301, −7.629074080892774, −7.194276881870006, −6.325285957790935, −5.847112942722987, −5.107345398009368, −4.692532708038561, −3.813870853444959, −3.274317914839049, −2.747956595298336, −1.687493020643439, −1.026348805650460, 0,
1.026348805650460, 1.687493020643439, 2.747956595298336, 3.274317914839049, 3.813870853444959, 4.692532708038561, 5.107345398009368, 5.847112942722987, 6.325285957790935, 7.194276881870006, 7.629074080892774, 7.975375580761301, 8.696336172288354, 9.451565062330952, 9.681658412610008, 10.43310899446187, 10.87995629751232, 11.60265227411908, 11.97782273269428, 12.39437039415526, 13.04594094125066, 13.55681294354865, 14.18403814363642, 14.66296920691594, 14.96518277248433
