L(s) = 1 | + 2.05·3-s − 1.81·5-s + 4.15·7-s + 1.23·9-s − 11-s + 1.63·13-s − 3.72·15-s − 7.92·17-s + 1.08·19-s + 8.55·21-s − 5.24·23-s − 1.71·25-s − 3.63·27-s − 1.03·29-s − 4.77·31-s − 2.05·33-s − 7.54·35-s − 4.73·37-s + 3.36·39-s − 5.23·41-s − 2.38·43-s − 2.23·45-s + 0.857·47-s + 10.2·49-s − 16.3·51-s − 7.45·53-s + 1.81·55-s + ⋯ |
L(s) = 1 | + 1.18·3-s − 0.810·5-s + 1.57·7-s + 0.410·9-s − 0.301·11-s + 0.453·13-s − 0.963·15-s − 1.92·17-s + 0.249·19-s + 1.86·21-s − 1.09·23-s − 0.342·25-s − 0.699·27-s − 0.191·29-s − 0.856·31-s − 0.358·33-s − 1.27·35-s − 0.777·37-s + 0.539·39-s − 0.817·41-s − 0.363·43-s − 0.333·45-s + 0.125·47-s + 1.47·49-s − 2.28·51-s − 1.02·53-s + 0.244·55-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 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 - 4.15T + 7T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 + 1.03T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 - 0.857T + 47T^{2} \) |
| 53 | \( 1 + 7.45T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 2.36T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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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
−7.990324871032719424159792372147, −7.36303765297260011985340385783, −6.45044842384153566500833488755, −5.40750683105531287242935995615, −4.60759669635863213011611918061, −4.00669806688918740935610662164, −3.30612239330462404127873507033, −2.15729857466261083380985984372, −1.74890346933612193361784153868, 0,
1.74890346933612193361784153868, 2.15729857466261083380985984372, 3.30612239330462404127873507033, 4.00669806688918740935610662164, 4.60759669635863213011611918061, 5.40750683105531287242935995615, 6.45044842384153566500833488755, 7.36303765297260011985340385783, 7.990324871032719424159792372147