| L(s) = 1 | + 3.09·5-s + 7·7-s − 26.0·11-s + 73.3·13-s − 75.2·17-s − 33.3·19-s + 49.3·23-s − 115.·25-s − 156.·29-s + 312.·31-s + 21.6·35-s − 120.·37-s − 260.·41-s − 177.·43-s − 186.·47-s + 49·49-s + 597.·53-s − 80.5·55-s − 811.·59-s − 168.·61-s + 226.·65-s − 220.·67-s + 621.·71-s − 346.·73-s − 182.·77-s − 193.·79-s + 481.·83-s + ⋯ |
| L(s) = 1 | + 0.276·5-s + 0.377·7-s − 0.714·11-s + 1.56·13-s − 1.07·17-s − 0.402·19-s + 0.447·23-s − 0.923·25-s − 1.00·29-s + 1.81·31-s + 0.104·35-s − 0.536·37-s − 0.991·41-s − 0.629·43-s − 0.578·47-s + 0.142·49-s + 1.54·53-s − 0.197·55-s − 1.78·59-s − 0.352·61-s + 0.432·65-s − 0.402·67-s + 1.03·71-s − 0.555·73-s − 0.270·77-s − 0.275·79-s + 0.637·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 3.09T + 125T^{2} \) |
| 11 | \( 1 + 26.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 312.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 260.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 177.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 597.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 811.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 168.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 220.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 621.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 346.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 193.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 386.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 286.T + 9.12e5T^{2} \) |
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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
−8.458803792027439921945830438603, −7.76450879491345987096192775980, −6.69356604333838615941513138281, −6.08629158879028098137316608699, −5.19435046336771040245853911180, −4.32878232798313819165198466408, −3.39673626556895228916261609569, −2.27750565039732330030004433511, −1.35989220530623636081230328782, 0,
1.35989220530623636081230328782, 2.27750565039732330030004433511, 3.39673626556895228916261609569, 4.32878232798313819165198466408, 5.19435046336771040245853911180, 6.08629158879028098137316608699, 6.69356604333838615941513138281, 7.76450879491345987096192775980, 8.458803792027439921945830438603
