L(s) = 1 | + 2.74·3-s + 2.04·5-s + 1.49·7-s + 4.53·9-s + 2.37·11-s + 2.66·13-s + 5.60·15-s + 0.879·17-s + 6.71·19-s + 4.11·21-s − 3.16·23-s − 0.831·25-s + 4.20·27-s + 1.61·29-s − 5.94·31-s + 6.51·33-s + 3.06·35-s − 8.58·37-s + 7.30·39-s + 10.7·41-s − 4.67·43-s + 9.25·45-s + 5.49·47-s − 4.75·49-s + 2.41·51-s − 7.37·53-s + 4.84·55-s + ⋯ |
L(s) = 1 | + 1.58·3-s + 0.913·5-s + 0.566·7-s + 1.51·9-s + 0.716·11-s + 0.738·13-s + 1.44·15-s + 0.213·17-s + 1.54·19-s + 0.897·21-s − 0.659·23-s − 0.166·25-s + 0.808·27-s + 0.300·29-s − 1.06·31-s + 1.13·33-s + 0.517·35-s − 1.41·37-s + 1.16·39-s + 1.67·41-s − 0.713·43-s + 1.37·45-s + 0.801·47-s − 0.678·49-s + 0.338·51-s − 1.01·53-s + 0.653·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.368840611\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.368840611\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 2.66T + 13T^{2} \) |
| 17 | \( 1 - 0.879T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 - 5.49T + 47T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 + 1.00T + 97T^{2} \) |
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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
−8.111241711609890049999992123286, −7.57566802629900758532471001172, −6.83536297266988482102750112571, −5.89402825807477481320381535528, −5.27429345213263463903958297980, −4.15667756630638624412384953067, −3.55369245847305201209183941474, −2.77992399727208271753661827833, −1.82287330156070356077130239307, −1.33847927448735934667822804687,
1.33847927448735934667822804687, 1.82287330156070356077130239307, 2.77992399727208271753661827833, 3.55369245847305201209183941474, 4.15667756630638624412384953067, 5.27429345213263463903958297980, 5.89402825807477481320381535528, 6.83536297266988482102750112571, 7.57566802629900758532471001172, 8.111241711609890049999992123286