| L(s) = 1 | + 32·2-s − 451.·3-s + 1.02e3·4-s − 1.44e4·6-s − 1.83e4·7-s + 3.27e4·8-s + 1.44e5·9-s − 7.32e4·11-s − 4.61e5·12-s − 7.42e5·13-s − 5.86e5·14-s + 1.04e6·16-s − 1.41e6·17-s + 4.62e6·18-s + 8.27e6·21-s − 2.34e6·22-s + 8.72e6·23-s − 1.47e7·24-s + 9.76e6·25-s − 2.37e7·26-s − 3.85e7·27-s − 1.87e7·28-s + 5.15e7·31-s + 3.35e7·32-s + 3.30e7·33-s − 4.54e7·34-s + 1.47e8·36-s + ⋯ |
| L(s) = 1 | + 2-s − 1.85·3-s + 4-s − 1.85·6-s − 1.09·7-s + 8-s + 2.44·9-s − 0.455·11-s − 1.85·12-s − 1.99·13-s − 1.09·14-s + 16-s − 0.999·17-s + 2.44·18-s + 2.02·21-s − 0.455·22-s + 1.35·23-s − 1.85·24-s + 25-s − 1.99·26-s − 2.68·27-s − 1.09·28-s + 1.79·31-s + 32-s + 0.844·33-s − 0.999·34-s + 2.44·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.395929242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395929242\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 17 | \( 1 + 1.41e6T \) |
| good | 3 | \( 1 + 451.T + 5.90e4T^{2} \) |
| 5 | \( 1 - 9.76e6T^{2} \) |
| 7 | \( 1 + 1.83e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + 7.32e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + 7.42e5T + 1.37e11T^{2} \) |
| 19 | \( 1 - 6.13e12T^{2} \) |
| 23 | \( 1 - 8.72e6T + 4.14e13T^{2} \) |
| 29 | \( 1 - 4.20e14T^{2} \) |
| 31 | \( 1 - 5.15e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 4.80e15T^{2} \) |
| 41 | \( 1 - 1.34e16T^{2} \) |
| 43 | \( 1 - 2.16e16T^{2} \) |
| 47 | \( 1 - 5.25e16T^{2} \) |
| 53 | \( 1 - 8.07e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 5.11e17T^{2} \) |
| 61 | \( 1 - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.82e18T^{2} \) |
| 71 | \( 1 - 3.58e9T + 3.25e18T^{2} \) |
| 73 | \( 1 - 4.29e18T^{2} \) |
| 79 | \( 1 + 5.96e9T + 9.46e18T^{2} \) |
| 83 | \( 1 - 1.55e19T^{2} \) |
| 89 | \( 1 - 8.54e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 7.37e19T^{2} \) |
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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
−12.57635832128367608277489824065, −11.81990826961108852546419966394, −10.71147455232044387013333683304, −9.870560953871413905444516873321, −7.11523251700564506902916567272, −6.56651524027178804588333518744, −5.27420941574072731511662012271, −4.54846684572329702367994008906, −2.63774117773464714304214533446, −0.62429920822363523484301381551,
0.62429920822363523484301381551, 2.63774117773464714304214533446, 4.54846684572329702367994008906, 5.27420941574072731511662012271, 6.56651524027178804588333518744, 7.11523251700564506902916567272, 9.870560953871413905444516873321, 10.71147455232044387013333683304, 11.81990826961108852546419966394, 12.57635832128367608277489824065
