L(s) = 1 | + 0.300·2-s + 0.482·3-s − 1.90·4-s + 5-s + 0.144·6-s + 5.09·7-s − 1.17·8-s − 2.76·9-s + 0.300·10-s + 0.862·11-s − 0.921·12-s + 3.43·13-s + 1.53·14-s + 0.482·15-s + 3.46·16-s − 0.818·17-s − 0.830·18-s + 2.57·19-s − 1.90·20-s + 2.45·21-s + 0.259·22-s − 5.66·23-s − 0.566·24-s + 25-s + 1.03·26-s − 2.78·27-s − 9.73·28-s + ⋯ |
L(s) = 1 | + 0.212·2-s + 0.278·3-s − 0.954·4-s + 0.447·5-s + 0.0591·6-s + 1.92·7-s − 0.415·8-s − 0.922·9-s + 0.0949·10-s + 0.260·11-s − 0.266·12-s + 0.951·13-s + 0.408·14-s + 0.124·15-s + 0.866·16-s − 0.198·17-s − 0.195·18-s + 0.590·19-s − 0.427·20-s + 0.536·21-s + 0.0552·22-s − 1.18·23-s − 0.115·24-s + 0.200·25-s + 0.202·26-s − 0.535·27-s − 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.100322885\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.100322885\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 0.300T + 2T^{2} \) |
| 3 | \( 1 - 0.482T + 3T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 - 0.862T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 + 0.818T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 - 7.33T + 41T^{2} \) |
| 43 | \( 1 - 9.02T + 43T^{2} \) |
| 47 | \( 1 + 2.26T + 47T^{2} \) |
| 53 | \( 1 - 0.408T + 53T^{2} \) |
| 59 | \( 1 - 1.97T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 - 0.108T + 71T^{2} \) |
| 73 | \( 1 + 2.86T + 73T^{2} \) |
| 79 | \( 1 - 6.71T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 - 7.99T + 97T^{2} \) |
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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
−9.515833078635080222277122432060, −8.835251962086153133298764799965, −8.213599263636690877540821312830, −7.65173268088040645145648837996, −5.99501268246962116911095733771, −5.52989829654323431542994946347, −4.54962982226761570015502308373, −3.80320702470249525508161975505, −2.39655632286010910517193298746, −1.14932162813680939973299944449,
1.14932162813680939973299944449, 2.39655632286010910517193298746, 3.80320702470249525508161975505, 4.54962982226761570015502308373, 5.52989829654323431542994946347, 5.99501268246962116911095733771, 7.65173268088040645145648837996, 8.213599263636690877540821312830, 8.835251962086153133298764799965, 9.515833078635080222277122432060