| L(s) = 1 | − 2.85·2-s − 8.98·3-s + 0.149·4-s + 25.6·6-s − 7·7-s + 22.4·8-s + 53.7·9-s + 37.4·11-s − 1.34·12-s − 3.96·13-s + 19.9·14-s − 65.1·16-s − 51.6·17-s − 153.·18-s + 25.9·19-s + 62.9·21-s − 106.·22-s + 173.·23-s − 201.·24-s + 11.3·26-s − 240.·27-s − 1.04·28-s − 245.·29-s − 172.·31-s + 6.76·32-s − 336.·33-s + 147.·34-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 1.72·3-s + 0.0186·4-s + 1.74·6-s − 0.377·7-s + 0.990·8-s + 1.99·9-s + 1.02·11-s − 0.0323·12-s − 0.0845·13-s + 0.381·14-s − 1.01·16-s − 0.737·17-s − 2.01·18-s + 0.313·19-s + 0.653·21-s − 1.03·22-s + 1.57·23-s − 1.71·24-s + 0.0853·26-s − 1.71·27-s − 0.00706·28-s − 1.57·29-s − 0.996·31-s + 0.0373·32-s − 1.77·33-s + 0.744·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 2.85T + 8T^{2} \) |
| 3 | \( 1 + 8.98T + 27T^{2} \) |
| 11 | \( 1 - 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.96T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 245.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 48.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 143.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 36.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 645.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 395.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 263.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 268.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 199.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 473.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 72.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 243.T + 9.12e5T^{2} \) |
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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
−11.28125826546917506493350277611, −10.96292159390626253631918404149, −9.701227383616021124175985670372, −9.060989595166397437626538911249, −7.39485695139970196937838556210, −6.58638271756813465217255796422, −5.35691336083601992329138299271, −4.18486505924493661207891435071, −1.28562818361079764931972099703, 0,
1.28562818361079764931972099703, 4.18486505924493661207891435071, 5.35691336083601992329138299271, 6.58638271756813465217255796422, 7.39485695139970196937838556210, 9.060989595166397437626538911249, 9.701227383616021124175985670372, 10.96292159390626253631918404149, 11.28125826546917506493350277611
