| L(s) = 1 | + 2.24e4·3-s + 1.95e6·5-s − 8.55e7·7-s − 6.59e8·9-s + 1.06e10·11-s + 2.87e9·13-s + 4.37e10·15-s + 1.15e11·17-s + 6.20e11·19-s − 1.91e12·21-s + 1.39e13·23-s + 3.81e12·25-s − 4.08e13·27-s − 2.09e13·29-s − 6.89e13·31-s + 2.38e14·33-s − 1.66e14·35-s + 1.76e14·37-s + 6.45e13·39-s + 3.64e15·41-s + 1.21e15·43-s − 1.28e15·45-s + 8.81e15·47-s − 4.08e15·49-s + 2.58e15·51-s + 3.00e16·53-s + 2.07e16·55-s + ⋯ |
| L(s) = 1 | + 0.657·3-s + 0.447·5-s − 0.800·7-s − 0.567·9-s + 1.35·11-s + 0.0753·13-s + 0.294·15-s + 0.235·17-s + 0.441·19-s − 0.526·21-s + 1.61·23-s + 0.199·25-s − 1.03·27-s − 0.268·29-s − 0.468·31-s + 0.892·33-s − 0.358·35-s + 0.223·37-s + 0.0495·39-s + 1.73·41-s + 0.368·43-s − 0.253·45-s + 1.14·47-s − 0.358·49-s + 0.154·51-s + 1.25·53-s + 0.607·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(2.767191988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.767191988\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 - 2.24e4T + 1.16e9T^{2} \) |
| 7 | \( 1 + 8.55e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.06e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 2.87e9T + 1.46e21T^{2} \) |
| 17 | \( 1 - 1.15e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 6.20e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.39e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 2.09e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 6.89e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.76e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.64e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 1.21e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 8.81e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.00e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 9.02e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 7.62e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.30e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.87e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.19e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.12e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 9.31e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 4.12e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.26e19T + 5.60e37T^{2} \) |
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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
−13.99635563856025831941366907249, −12.79462259399594142338750540938, −11.30200975411589931771154774260, −9.586123589318187869483352529720, −8.827102813935929252885361751903, −7.05528201375528510111916213798, −5.72581807295043829336424337061, −3.75923506411813803371207855671, −2.59780546963172511451766996882, −0.959767994095335371099248927992,
0.959767994095335371099248927992, 2.59780546963172511451766996882, 3.75923506411813803371207855671, 5.72581807295043829336424337061, 7.05528201375528510111916213798, 8.827102813935929252885361751903, 9.586123589318187869483352529720, 11.30200975411589931771154774260, 12.79462259399594142338750540938, 13.99635563856025831941366907249
