| L(s) = 1 | − 2.25·2-s + 3.09·4-s − 1.69·5-s + 0.963·7-s − 2.47·8-s + 3.83·10-s − 11-s − 4.15·13-s − 2.17·14-s − 0.611·16-s − 6.69·17-s + 1.25·19-s − 5.25·20-s + 2.25·22-s − 0.411·23-s − 2.11·25-s + 9.38·26-s + 2.98·28-s + 3.52·29-s + 3.52·31-s + 6.32·32-s + 15.1·34-s − 1.63·35-s + 11.0·37-s − 2.82·38-s + 4.19·40-s + 8.91·41-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.54·4-s − 0.759·5-s + 0.364·7-s − 0.873·8-s + 1.21·10-s − 0.301·11-s − 1.15·13-s − 0.581·14-s − 0.152·16-s − 1.62·17-s + 0.287·19-s − 1.17·20-s + 0.481·22-s − 0.0857·23-s − 0.423·25-s + 1.84·26-s + 0.563·28-s + 0.655·29-s + 0.632·31-s + 1.11·32-s + 2.59·34-s − 0.276·35-s + 1.80·37-s − 0.458·38-s + 0.663·40-s + 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 - 0.963T + 7T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 + 6.69T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 0.411T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 8.91T + 41T^{2} \) |
| 43 | \( 1 - 6.77T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 - 0.321T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 9.11T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 + 0.431T + 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−7.913973685323588561564159852421, −7.31098244432610910636880321905, −6.71613333202502837754161711249, −5.80084287101460557264343192897, −4.58566641048904861121180716455, −4.25396884685013090462940734152, −2.70277534931372617956834249922, −2.24537652181271127555958974599, −0.946906606420737313287241053218, 0,
0.946906606420737313287241053218, 2.24537652181271127555958974599, 2.70277534931372617956834249922, 4.25396884685013090462940734152, 4.58566641048904861121180716455, 5.80084287101460557264343192897, 6.71613333202502837754161711249, 7.31098244432610910636880321905, 7.913973685323588561564159852421
