| L(s) = 1 | + 56.4·2-s + 2.11e3·3-s − 5.00e3·4-s + 1.56e4·5-s + 1.19e5·6-s + 3.25e5·7-s − 7.44e5·8-s + 2.87e6·9-s + 8.81e5·10-s − 1.61e6·11-s − 1.05e7·12-s − 3.19e7·13-s + 1.83e7·14-s + 3.30e7·15-s − 9.98e5·16-s + 3.82e6·17-s + 1.62e8·18-s − 1.98e8·19-s − 7.82e7·20-s + 6.88e8·21-s − 9.10e7·22-s − 1.86e8·23-s − 1.57e9·24-s + 2.44e8·25-s − 1.80e9·26-s + 2.71e9·27-s − 1.62e9·28-s + ⋯ |
| L(s) = 1 | + 0.623·2-s + 1.67·3-s − 0.611·4-s + 0.447·5-s + 1.04·6-s + 1.04·7-s − 1.00·8-s + 1.80·9-s + 0.278·10-s − 0.274·11-s − 1.02·12-s − 1.83·13-s + 0.651·14-s + 0.749·15-s − 0.0148·16-s + 0.0384·17-s + 1.12·18-s − 0.969·19-s − 0.273·20-s + 1.75·21-s − 0.171·22-s − 0.262·23-s − 1.68·24-s + 0.199·25-s − 1.14·26-s + 1.34·27-s − 0.638·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.013951144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013951144\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.56e4T \) |
| good | 2 | \( 1 - 56.4T + 8.19e3T^{2} \) |
| 3 | \( 1 - 2.11e3T + 1.59e6T^{2} \) |
| 7 | \( 1 - 3.25e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 1.61e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.19e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 3.82e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.98e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.86e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.45e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 9.66e8T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.20e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.05e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.28e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 7.97e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.25e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 7.96e10T + 1.04e23T^{2} \) |
| 61 | \( 1 - 4.91e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 2.25e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 6.50e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.03e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.08e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.39e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.20e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.00e12T + 6.73e25T^{2} \) |
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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
−20.90614686498054961947325615198, −19.36569272444477353779189042493, −17.74095729215255639128039610725, −14.80976008223161529741057830248, −14.28036333110262779137993025988, −12.80579314586801838878171470279, −9.578739638157912096536922881757, −8.053295773867489441672942657162, −4.58968133485545444703984261478, −2.45800016280688572336685159833,
2.45800016280688572336685159833, 4.58968133485545444703984261478, 8.053295773867489441672942657162, 9.578739638157912096536922881757, 12.80579314586801838878171470279, 14.28036333110262779137993025988, 14.80976008223161529741057830248, 17.74095729215255639128039610725, 19.36569272444477353779189042493, 20.90614686498054961947325615198
