| L(s) = 1 | + 19.5·5-s + 6.26·7-s − 27.2·11-s − 13·13-s − 30.8·17-s − 127.·19-s + 84.3·23-s + 256.·25-s − 272.·29-s + 166.·31-s + 122.·35-s − 198.·37-s − 160.·41-s − 158.·43-s − 305.·47-s − 303.·49-s − 356.·53-s − 532.·55-s − 470.·59-s − 171.·61-s − 253.·65-s − 1.02e3·67-s − 188.·71-s + 959.·73-s − 170.·77-s + 1.03e3·79-s − 105.·83-s + ⋯ |
| L(s) = 1 | + 1.74·5-s + 0.338·7-s − 0.746·11-s − 0.277·13-s − 0.440·17-s − 1.54·19-s + 0.764·23-s + 2.05·25-s − 1.74·29-s + 0.963·31-s + 0.590·35-s − 0.880·37-s − 0.610·41-s − 0.563·43-s − 0.949·47-s − 0.885·49-s − 0.924·53-s − 1.30·55-s − 1.03·59-s − 0.360·61-s − 0.484·65-s − 1.86·67-s − 0.315·71-s + 1.53·73-s − 0.252·77-s + 1.47·79-s − 0.139·83-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 | \( 1 - 19.5T + 125T^{2} \) |
| 7 | \( 1 - 6.26T + 343T^{2} \) |
| 11 | \( 1 + 27.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 30.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 470.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 188.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 959.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 105.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 649.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 707.T + 9.12e5T^{2} \) |
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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
−8.646623914611834586257100924701, −7.73164030753458184931766418099, −6.65191691000371093910586536975, −6.14966468825330265178899740107, −5.18687656309643816127039088363, −4.69043161417104098283509897995, −3.19220775814200004406469152421, −2.17587970201688029506931360778, −1.62794123582672902760398543752, 0,
1.62794123582672902760398543752, 2.17587970201688029506931360778, 3.19220775814200004406469152421, 4.69043161417104098283509897995, 5.18687656309643816127039088363, 6.14966468825330265178899740107, 6.65191691000371093910586536975, 7.73164030753458184931766418099, 8.646623914611834586257100924701
