| L(s) = 1 | + 915.·2-s − 1.25e6·4-s + 9.76e6·5-s − 9.67e8·7-s − 3.07e9·8-s + 8.93e9·10-s − 5.85e10·11-s − 8.35e11·13-s − 8.85e11·14-s − 1.70e11·16-s + 2.88e12·17-s − 1.26e13·19-s − 1.22e13·20-s − 5.35e13·22-s − 3.44e14·23-s + 9.53e13·25-s − 7.64e14·26-s + 1.21e15·28-s + 1.05e15·29-s + 2.70e15·31-s + 6.28e15·32-s + 2.63e15·34-s − 9.44e15·35-s + 3.58e15·37-s − 1.15e16·38-s − 3.00e16·40-s − 8.22e16·41-s + ⋯ |
| L(s) = 1 | + 0.632·2-s − 0.600·4-s + 0.447·5-s − 1.29·7-s − 1.01·8-s + 0.282·10-s − 0.680·11-s − 1.68·13-s − 0.817·14-s − 0.0387·16-s + 0.346·17-s − 0.472·19-s − 0.268·20-s − 0.430·22-s − 1.73·23-s + 0.199·25-s − 1.06·26-s + 0.777·28-s + 0.463·29-s + 0.593·31-s + 0.987·32-s + 0.219·34-s − 0.578·35-s + 0.122·37-s − 0.298·38-s − 0.452·40-s − 0.957·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.6420705665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6420705665\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 9.76e6T \) |
| good | 2 | \( 1 - 915.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 9.67e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 5.85e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 8.35e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.88e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.26e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.44e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.05e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.70e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.58e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 8.22e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 3.16e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.88e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.17e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 3.50e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.82e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.95e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.20e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.95e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.49e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 8.63e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 8.04e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.75e19T + 5.27e41T^{2} \) |
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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
−12.11696555862616118892751642484, −10.03405821024484505312345503390, −9.690026108396479406147420348827, −8.176001435934996699323925508595, −6.64622803264921020552744809936, −5.60917509226427980368860414391, −4.56489281927759100036223554661, −3.28013056562113749404616411189, −2.32022928145200749591629223353, −0.31336699702421891306145995900,
0.31336699702421891306145995900, 2.32022928145200749591629223353, 3.28013056562113749404616411189, 4.56489281927759100036223554661, 5.60917509226427980368860414391, 6.64622803264921020552744809936, 8.176001435934996699323925508595, 9.690026108396479406147420348827, 10.03405821024484505312345503390, 12.11696555862616118892751642484
