| L(s) = 1 | − 1.52·2-s + 9.11·3-s − 5.68·4-s − 13.8·6-s − 16.3·7-s + 20.8·8-s + 56.1·9-s − 25.6·11-s − 51.8·12-s − 52.2·13-s + 24.8·14-s + 13.8·16-s − 17·17-s − 85.3·18-s + 47.4·19-s − 149.·21-s + 38.9·22-s − 43.0·23-s + 189.·24-s + 79.4·26-s + 265.·27-s + 93.0·28-s − 135.·29-s − 302.·31-s − 187.·32-s − 233.·33-s + 25.8·34-s + ⋯ |
| L(s) = 1 | − 0.537·2-s + 1.75·3-s − 0.711·4-s − 0.943·6-s − 0.883·7-s + 0.919·8-s + 2.07·9-s − 0.702·11-s − 1.24·12-s − 1.11·13-s + 0.474·14-s + 0.216·16-s − 0.242·17-s − 1.11·18-s + 0.573·19-s − 1.55·21-s + 0.377·22-s − 0.390·23-s + 1.61·24-s + 0.599·26-s + 1.89·27-s + 0.628·28-s − 0.864·29-s − 1.75·31-s − 1.03·32-s − 1.23·33-s + 0.130·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 3 | \( 1 - 9.11T + 27T^{2} \) |
| 7 | \( 1 + 16.3T + 343T^{2} \) |
| 11 | \( 1 + 25.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 47.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 302.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 159.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 562.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 24.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 430.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 105.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 334.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 751.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 845.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 106.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 35.8T + 9.12e5T^{2} \) |
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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
−9.703895399997746960200091565743, −9.493077125755260633712220078608, −8.616897210124770141184253135152, −7.69567930141038720470215588789, −7.16334659302847444532257739973, −5.30207028857165183710073217305, −4.03643332916246655663454454925, −3.10865382454852942656713926570, −1.93149043706285290329073491493, 0,
1.93149043706285290329073491493, 3.10865382454852942656713926570, 4.03643332916246655663454454925, 5.30207028857165183710073217305, 7.16334659302847444532257739973, 7.69567930141038720470215588789, 8.616897210124770141184253135152, 9.493077125755260633712220078608, 9.703895399997746960200091565743
