| L(s) = 1 | − 8·2-s + 38.5·3-s + 64·4-s − 399.·5-s − 308.·6-s − 343·7-s − 512·8-s − 702.·9-s + 3.19e3·10-s + 4.09e3·11-s + 2.46e3·12-s + 2.19e3·13-s + 2.74e3·14-s − 1.53e4·15-s + 4.09e3·16-s + 3.67e4·17-s + 5.62e3·18-s + 3.00e4·19-s − 2.55e4·20-s − 1.32e4·21-s − 3.27e4·22-s + 5.92e4·23-s − 1.97e4·24-s + 8.13e4·25-s − 1.75e4·26-s − 1.11e5·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.823·3-s + 0.5·4-s − 1.42·5-s − 0.582·6-s − 0.377·7-s − 0.353·8-s − 0.321·9-s + 1.01·10-s + 0.928·11-s + 0.411·12-s + 0.277·13-s + 0.267·14-s − 1.17·15-s + 0.250·16-s + 1.81·17-s + 0.227·18-s + 1.00·19-s − 0.714·20-s − 0.311·21-s − 0.656·22-s + 1.01·23-s − 0.291·24-s + 1.04·25-s − 0.196·26-s − 1.08·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 3 | \( 1 - 38.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 399.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.09e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.00e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.92e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.60e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.96e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.21e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.87e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.65e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.73e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.53e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.69e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.63e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.79e6T + 8.07e13T^{2} \) |
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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
−10.91259348325668762282103088117, −9.461293300898360588955819960278, −8.882389482038442540099036711661, −7.73142629625884356139318992869, −7.29426465389286863428830888695, −5.62993640620890476045633453348, −3.66708138787889528881242327257, −3.22360644963647464885642298803, −1.34827494897344021267745164536, 0,
1.34827494897344021267745164536, 3.22360644963647464885642298803, 3.66708138787889528881242327257, 5.62993640620890476045633453348, 7.29426465389286863428830888695, 7.73142629625884356139318992869, 8.882389482038442540099036711661, 9.461293300898360588955819960278, 10.91259348325668762282103088117
