| L(s) = 1 | − 1.65e3·2-s − 4.81e6·3-s − 1.31e8·4-s + 1.22e9·5-s + 7.95e9·6-s + 4.20e11·7-s + 4.38e11·8-s + 1.55e13·9-s − 2.01e12·10-s − 8.09e13·11-s + 6.33e14·12-s − 1.61e15·13-s − 6.94e14·14-s − 5.87e15·15-s + 1.69e16·16-s + 2.88e16·17-s − 2.57e16·18-s + 1.06e17·19-s − 1.60e17·20-s − 2.02e18·21-s + 1.33e17·22-s + 1.88e18·23-s − 2.11e18·24-s + 1.49e18·25-s + 2.67e18·26-s − 3.82e19·27-s − 5.52e19·28-s + ⋯ |
| L(s) = 1 | − 0.142·2-s − 1.74·3-s − 0.979·4-s + 0.447·5-s + 0.248·6-s + 1.63·7-s + 0.282·8-s + 2.04·9-s − 0.0637·10-s − 0.707·11-s + 1.70·12-s − 1.48·13-s − 0.233·14-s − 0.780·15-s + 0.939·16-s + 0.707·17-s − 0.291·18-s + 0.582·19-s − 0.438·20-s − 2.86·21-s + 0.100·22-s + 0.778·23-s − 0.492·24-s + 0.199·25-s + 0.211·26-s − 1.81·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.22e9T \) |
| good | 2 | \( 1 + 1.65e3T + 1.34e8T^{2} \) |
| 3 | \( 1 + 4.81e6T + 7.62e12T^{2} \) |
| 7 | \( 1 - 4.20e11T + 6.57e22T^{2} \) |
| 11 | \( 1 + 8.09e13T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.61e15T + 1.19e30T^{2} \) |
| 17 | \( 1 - 2.88e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 1.06e17T + 3.36e34T^{2} \) |
| 23 | \( 1 - 1.88e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 3.26e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 1.12e20T + 1.84e40T^{2} \) |
| 37 | \( 1 - 2.79e20T + 2.19e42T^{2} \) |
| 41 | \( 1 - 3.98e21T + 3.50e43T^{2} \) |
| 43 | \( 1 + 1.18e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 5.82e22T + 1.40e45T^{2} \) |
| 53 | \( 1 + 3.63e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 1.26e23T + 6.50e47T^{2} \) |
| 61 | \( 1 - 9.53e23T + 1.59e48T^{2} \) |
| 67 | \( 1 + 1.65e23T + 2.01e49T^{2} \) |
| 71 | \( 1 - 5.68e24T + 9.63e49T^{2} \) |
| 73 | \( 1 - 4.04e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 2.07e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 1.08e26T + 6.53e51T^{2} \) |
| 89 | \( 1 - 3.61e25T + 4.30e52T^{2} \) |
| 97 | \( 1 + 6.53e26T + 4.39e53T^{2} \) |
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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
−16.86214055403660771395499253843, −14.56246993739230668936109437079, −12.69278402890317700240313179630, −11.25507282812652510678807777242, −9.898611133528876851345967820080, −7.62982015484408550703464721746, −5.29942206780169066722718645992, −4.87575117703384413375490745794, −1.37609273198182441622915377216, 0,
1.37609273198182441622915377216, 4.87575117703384413375490745794, 5.29942206780169066722718645992, 7.62982015484408550703464721746, 9.898611133528876851345967820080, 11.25507282812652510678807777242, 12.69278402890317700240313179630, 14.56246993739230668936109437079, 16.86214055403660771395499253843
