| L(s) = 1 | − 1.46·5-s + 8.39·7-s + 34.7·11-s − 13·13-s + 108.·17-s − 143.·19-s − 128.·23-s − 122.·25-s + 18.8·29-s + 78.5·31-s − 12.2·35-s − 327.·37-s − 327.·41-s + 336.·43-s + 99.2·47-s − 272.·49-s + 686.·53-s − 50.9·55-s − 242.·59-s − 644.·61-s + 19.0·65-s + 871.·67-s + 100.·71-s + 604.·73-s + 291.·77-s − 1.07e3·79-s + 741.·83-s + ⋯ |
| L(s) = 1 | − 0.130·5-s + 0.453·7-s + 0.953·11-s − 0.277·13-s + 1.54·17-s − 1.72·19-s − 1.16·23-s − 0.982·25-s + 0.120·29-s + 0.455·31-s − 0.0593·35-s − 1.45·37-s − 1.24·41-s + 1.19·43-s + 0.308·47-s − 0.794·49-s + 1.77·53-s − 0.124·55-s − 0.534·59-s − 1.35·61-s + 0.0363·65-s + 1.58·67-s + 0.167·71-s + 0.969·73-s + 0.432·77-s − 1.52·79-s + 0.980·83-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 1.46T + 125T^{2} \) |
| 7 | \( 1 - 8.39T + 343T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 99.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 644.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 871.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 604.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 741.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 501.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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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
−8.360086228596442890697524805437, −7.85336744706432291335002852031, −6.86514895117093578021830889653, −6.10307184766743378180356063655, −5.26864437116058283106768035677, −4.20642827090801170623272426712, −3.62038399299676803120515399977, −2.26083533964262791808071566392, −1.36031215057164922009612106105, 0,
1.36031215057164922009612106105, 2.26083533964262791808071566392, 3.62038399299676803120515399977, 4.20642827090801170623272426712, 5.26864437116058283106768035677, 6.10307184766743378180356063655, 6.86514895117093578021830889653, 7.85336744706432291335002852031, 8.360086228596442890697524805437
