L(s) = 1 | + 2.59·3-s − 2.40·5-s + 0.934·7-s + 3.73·9-s + 11-s − 3.65·13-s − 6.24·15-s − 0.402·17-s − 2.79·19-s + 2.42·21-s − 3.78·23-s + 0.798·25-s + 1.89·27-s − 1.66·29-s + 9.72·31-s + 2.59·33-s − 2.25·35-s − 3.96·37-s − 9.48·39-s − 4.88·41-s + 0.416·43-s − 8.98·45-s + 7.36·47-s − 6.12·49-s − 1.04·51-s − 5.18·53-s − 2.40·55-s + ⋯ |
L(s) = 1 | + 1.49·3-s − 1.07·5-s + 0.353·7-s + 1.24·9-s + 0.301·11-s − 1.01·13-s − 1.61·15-s − 0.0975·17-s − 0.641·19-s + 0.529·21-s − 0.790·23-s + 0.159·25-s + 0.365·27-s − 0.309·29-s + 1.74·31-s + 0.451·33-s − 0.380·35-s − 0.651·37-s − 1.51·39-s − 0.763·41-s + 0.0634·43-s − 1.33·45-s + 1.07·47-s − 0.875·49-s − 0.146·51-s − 0.711·53-s − 0.324·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
good | 3 | \( 1 - 2.59T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 0.934T + 7T^{2} \) |
| 13 | \( 1 + 3.65T + 13T^{2} \) |
| 17 | \( 1 + 0.402T + 17T^{2} \) |
| 19 | \( 1 + 2.79T + 19T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 - 9.72T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 - 0.416T + 43T^{2} \) |
| 47 | \( 1 - 7.36T + 47T^{2} \) |
| 53 | \( 1 + 5.18T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 0.664T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.979135666140348655287838344598, −7.24691167592234184190471966013, −6.62803805447773164660160837465, −5.45452308941806980675998932621, −4.35200835002734064808111890948, −4.12691088963562440304927673889, −3.15564871879322347369459473675, −2.49331099578865169713908263051, −1.58495084410383424361514845478, 0,
1.58495084410383424361514845478, 2.49331099578865169713908263051, 3.15564871879322347369459473675, 4.12691088963562440304927673889, 4.35200835002734064808111890948, 5.45452308941806980675998932621, 6.62803805447773164660160837465, 7.24691167592234184190471966013, 7.979135666140348655287838344598