| L(s) = 1 | + 5·5-s − 7.95·7-s − 37.1·11-s − 35.1·13-s − 99.1·17-s − 44.6·19-s + 102.·23-s + 25·25-s + 285.·29-s + 238.·31-s − 39.7·35-s + 339.·37-s − 423.·41-s + 144.·43-s + 418.·47-s − 279.·49-s + 186.·53-s − 185.·55-s + 293.·59-s − 701.·61-s − 175.·65-s − 292.·67-s + 738.·71-s + 453.·73-s + 295.·77-s + 892.·79-s − 66.4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.429·7-s − 1.01·11-s − 0.749·13-s − 1.41·17-s − 0.539·19-s + 0.930·23-s + 0.200·25-s + 1.82·29-s + 1.38·31-s − 0.192·35-s + 1.50·37-s − 1.61·41-s + 0.511·43-s + 1.29·47-s − 0.815·49-s + 0.484·53-s − 0.455·55-s + 0.648·59-s − 1.47·61-s − 0.335·65-s − 0.533·67-s + 1.23·71-s + 0.726·73-s + 0.436·77-s + 1.27·79-s − 0.0878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.630840994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630840994\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 7.95T + 343T^{2} \) |
| 11 | \( 1 + 37.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 423.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 186.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 293.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 292.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 738.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 453.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 892.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 66.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 868.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 112.T + 9.12e5T^{2} \) |
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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
−9.550082073370850545946615347114, −8.702558664098854338935417197135, −7.912820578663465604694880104542, −6.81228281417323192237787926239, −6.28350199026216973636423694460, −5.06202635912384229143057161145, −4.45924689905156759958639343959, −2.90730867745021272077376992361, −2.29917349529856224228811743399, −0.64426031929350432934824858120,
0.64426031929350432934824858120, 2.29917349529856224228811743399, 2.90730867745021272077376992361, 4.45924689905156759958639343959, 5.06202635912384229143057161145, 6.28350199026216973636423694460, 6.81228281417323192237787926239, 7.912820578663465604694880104542, 8.702558664098854338935417197135, 9.550082073370850545946615347114
