L(s) = 1 | + 1.21·2-s − 0.525·4-s − 1.31·5-s + 1.52·7-s − 3.06·8-s − 1.59·10-s + 11-s − 13-s + 1.85·14-s − 2.67·16-s − 2.21·17-s − 1.52·19-s + 0.688·20-s + 1.21·22-s − 7.95·23-s − 3.28·25-s − 1.21·26-s − 0.801·28-s − 7.39·29-s + 4.68·31-s + 2.88·32-s − 2.68·34-s − 1.99·35-s + 8.85·37-s − 1.85·38-s + 4.02·40-s + 3.52·41-s + ⋯ |
L(s) = 1 | + 0.858·2-s − 0.262·4-s − 0.586·5-s + 0.576·7-s − 1.08·8-s − 0.503·10-s + 0.301·11-s − 0.277·13-s + 0.495·14-s − 0.668·16-s − 0.537·17-s − 0.349·19-s + 0.154·20-s + 0.258·22-s − 1.65·23-s − 0.656·25-s − 0.238·26-s − 0.151·28-s − 1.37·29-s + 0.842·31-s + 0.510·32-s − 0.461·34-s − 0.338·35-s + 1.45·37-s − 0.300·38-s + 0.635·40-s + 0.550·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 17 | \( 1 + 2.21T + 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 + 7.39T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 8.77T + 43T^{2} \) |
| 47 | \( 1 + 9.18T + 47T^{2} \) |
| 53 | \( 1 + 3.67T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 5.25T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.13T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 + 0.688T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−9.323054900388373551835573610593, −8.231071929764411414763074333113, −7.82330534887788368921168340503, −6.51578926432790503620667146410, −5.83007554139145401003412558934, −4.72516249049651619751623232795, −4.22232489463179257214053116419, −3.33003828748902688356583089143, −1.98338631630392899987616614803, 0,
1.98338631630392899987616614803, 3.33003828748902688356583089143, 4.22232489463179257214053116419, 4.72516249049651619751623232795, 5.83007554139145401003412558934, 6.51578926432790503620667146410, 7.82330534887788368921168340503, 8.231071929764411414763074333113, 9.323054900388373551835573610593