| L(s) = 1 | + 2.17e4·2-s − 1.59e6·3-s + 3.36e8·4-s − 6.93e8·5-s − 3.46e10·6-s + 3.81e11·7-s + 4.39e12·8-s + 2.54e12·9-s − 1.50e13·10-s + 1.26e14·11-s − 5.36e14·12-s − 1.31e15·13-s + 8.27e15·14-s + 1.10e15·15-s + 5.02e16·16-s − 3.53e16·17-s + 5.51e16·18-s + 8.65e16·19-s − 2.33e17·20-s − 6.07e17·21-s + 2.74e18·22-s + 5.10e17·23-s − 7.00e18·24-s − 6.96e18·25-s − 2.85e19·26-s − 4.05e18·27-s + 1.28e20·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 0.577·3-s + 2.50·4-s − 0.254·5-s − 1.08·6-s + 1.48·7-s + 2.82·8-s + 0.333·9-s − 0.476·10-s + 1.10·11-s − 1.44·12-s − 1.20·13-s + 2.78·14-s + 0.146·15-s + 2.78·16-s − 0.865·17-s + 0.624·18-s + 0.471·19-s − 0.637·20-s − 0.858·21-s + 2.06·22-s + 0.211·23-s − 1.63·24-s − 0.935·25-s − 2.25·26-s − 0.192·27-s + 3.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(\approx\) |
\(5.058396498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.058396498\) |
| \(L(\frac{29}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.59e6T \) |
| good | 2 | \( 1 - 2.17e4T + 1.34e8T^{2} \) |
| 5 | \( 1 + 6.93e8T + 7.45e18T^{2} \) |
| 7 | \( 1 - 3.81e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 1.26e14T + 1.31e28T^{2} \) |
| 13 | \( 1 + 1.31e15T + 1.19e30T^{2} \) |
| 17 | \( 1 + 3.53e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 8.65e16T + 3.36e34T^{2} \) |
| 23 | \( 1 - 5.10e17T + 5.84e36T^{2} \) |
| 29 | \( 1 + 2.74e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 9.77e19T + 1.84e40T^{2} \) |
| 37 | \( 1 + 1.80e21T + 2.19e42T^{2} \) |
| 41 | \( 1 + 5.17e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 8.74e21T + 1.26e44T^{2} \) |
| 47 | \( 1 - 2.51e22T + 1.40e45T^{2} \) |
| 53 | \( 1 - 2.11e23T + 3.59e46T^{2} \) |
| 59 | \( 1 + 7.00e23T + 6.50e47T^{2} \) |
| 61 | \( 1 - 1.08e23T + 1.59e48T^{2} \) |
| 67 | \( 1 - 2.46e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.29e25T + 9.63e49T^{2} \) |
| 73 | \( 1 - 7.46e24T + 2.04e50T^{2} \) |
| 79 | \( 1 + 4.94e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 5.54e25T + 6.53e51T^{2} \) |
| 89 | \( 1 + 1.32e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 1.06e27T + 4.39e53T^{2} \) |
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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
−20.03702430900785282319792107584, −17.16706581868042150940126326892, −15.24382046246837735142537072943, −14.08355488248378304772540539740, −12.13461840817885297569941909304, −11.22625307050688578883812168274, −7.15881120904364726585389400392, −5.30940080005576982773030934164, −4.15308581087043677264637868676, −1.86461830146449799650916188094,
1.86461830146449799650916188094, 4.15308581087043677264637868676, 5.30940080005576982773030934164, 7.15881120904364726585389400392, 11.22625307050688578883812168274, 12.13461840817885297569941909304, 14.08355488248378304772540539740, 15.24382046246837735142537072943, 17.16706581868042150940126326892, 20.03702430900785282319792107584
