| L(s) = 1 | − 4.23·2-s + 3·3-s + 9.94·4-s − 5·5-s − 12.7·6-s − 7·7-s − 8.23·8-s + 9·9-s + 21.1·10-s − 41.5·11-s + 29.8·12-s + 88.9·13-s + 29.6·14-s − 15·15-s − 44.6·16-s − 120.·17-s − 38.1·18-s − 112.·19-s − 49.7·20-s − 21·21-s + 175.·22-s − 115.·23-s − 24.7·24-s + 25·25-s − 376.·26-s + 27·27-s − 69.6·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.24·4-s − 0.447·5-s − 0.864·6-s − 0.377·7-s − 0.363·8-s + 0.333·9-s + 0.669·10-s − 1.13·11-s + 0.717·12-s + 1.89·13-s + 0.566·14-s − 0.258·15-s − 0.697·16-s − 1.71·17-s − 0.499·18-s − 1.35·19-s − 0.555·20-s − 0.218·21-s + 1.70·22-s − 1.04·23-s − 0.210·24-s + 0.200·25-s − 2.84·26-s + 0.192·27-s − 0.469·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 4.23T + 8T^{2} \) |
| 11 | \( 1 + 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 48.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 575.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 529.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.28T + 3.00e5T^{2} \) |
| 71 | \( 1 + 61.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 878.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 491.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 415.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−12.88181569707492296853043281622, −11.02195019693420594559702105497, −10.65386272950849421210188056087, −9.168981172629181272824934474395, −8.535523732055619904725233737761, −7.62799052378416580530155802593, −6.33003777724750842570572638412, −4.01589893874121749634155343159, −2.09774235159822270881569792749, 0,
2.09774235159822270881569792749, 4.01589893874121749634155343159, 6.33003777724750842570572638412, 7.62799052378416580530155802593, 8.535523732055619904725233737761, 9.168981172629181272824934474395, 10.65386272950849421210188056087, 11.02195019693420594559702105497, 12.88181569707492296853043281622
