| L(s) = 1 | + 13.3·5-s − 15.3·7-s + 24.9·11-s + 13·13-s + 65.5·17-s + 73.1·19-s + 28.5·23-s + 54.0·25-s + 220.·29-s − 138.·31-s − 205.·35-s − 354.·37-s + 297.·41-s + 9.62·43-s − 219.·47-s − 106.·49-s + 189.·53-s + 333.·55-s + 329.·59-s − 838.·61-s + 173.·65-s + 386.·67-s + 664.·71-s + 248.·73-s − 383.·77-s + 1.26e3·79-s − 157.·83-s + ⋯ |
| L(s) = 1 | + 1.19·5-s − 0.830·7-s + 0.682·11-s + 0.277·13-s + 0.934·17-s + 0.883·19-s + 0.259·23-s + 0.432·25-s + 1.41·29-s − 0.805·31-s − 0.993·35-s − 1.57·37-s + 1.13·41-s + 0.0341·43-s − 0.681·47-s − 0.310·49-s + 0.490·53-s + 0.816·55-s + 0.726·59-s − 1.76·61-s + 0.331·65-s + 0.705·67-s + 1.11·71-s + 0.398·73-s − 0.566·77-s + 1.80·79-s − 0.208·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.968394989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968394989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 13.3T + 125T^{2} \) |
| 7 | \( 1 + 15.3T + 343T^{2} \) |
| 11 | \( 1 - 24.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 354.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 9.62T + 7.95e4T^{2} \) |
| 47 | \( 1 + 219.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 189.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 329.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 838.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 386.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 664.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 248.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 157.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−9.145802262897093197403581091646, −8.164651389729053288420877211121, −7.10061831436141253845213338292, −6.43037197773822141927867403422, −5.74763535301356381520756427143, −5.00736294209181984914737642352, −3.69366743423158997652602608756, −2.96834246161620011156434193081, −1.79743085641532931659034312279, −0.837844445562178959551874520853,
0.837844445562178959551874520853, 1.79743085641532931659034312279, 2.96834246161620011156434193081, 3.69366743423158997652602608756, 5.00736294209181984914737642352, 5.74763535301356381520756427143, 6.43037197773822141927867403422, 7.10061831436141253845213338292, 8.164651389729053288420877211121, 9.145802262897093197403581091646
