| L(s) = 1 | − 7.08·2-s + 27·3-s − 77.7·4-s − 194.·5-s − 191.·6-s − 182.·7-s + 1.45e3·8-s + 729·9-s + 1.37e3·10-s − 2.10e3·12-s + 7.08e3·13-s + 1.29e3·14-s − 5.24e3·15-s − 373.·16-s − 960.·17-s − 5.16e3·18-s + 1.89e4·19-s + 1.51e4·20-s − 4.93e3·21-s − 1.31e4·23-s + 3.93e4·24-s − 4.03e4·25-s − 5.02e4·26-s + 1.96e4·27-s + 1.42e4·28-s − 5.31e4·29-s + 3.71e4·30-s + ⋯ |
| L(s) = 1 | − 0.626·2-s + 0.577·3-s − 0.607·4-s − 0.695·5-s − 0.361·6-s − 0.201·7-s + 1.00·8-s + 0.333·9-s + 0.435·10-s − 0.350·12-s + 0.894·13-s + 0.126·14-s − 0.401·15-s − 0.0227·16-s − 0.0474·17-s − 0.208·18-s + 0.632·19-s + 0.422·20-s − 0.116·21-s − 0.225·23-s + 0.581·24-s − 0.516·25-s − 0.560·26-s + 0.192·27-s + 0.122·28-s − 0.404·29-s + 0.251·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.157141502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157141502\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 7.08T + 128T^{2} \) |
| 5 | \( 1 + 194.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 182.T + 8.23e5T^{2} \) |
| 13 | \( 1 - 7.08e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 960.T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.89e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.31e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.31e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.32e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.97e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.29e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.52e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.13e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.35e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.64e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.60e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.59e6T + 8.07e13T^{2} \) |
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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
−9.992323951065508470229482929495, −9.244829370886954283150748800787, −8.336417309746932619673485196296, −7.83487879245742495409687790477, −6.72294132753306925612716542557, −5.26469628943002324733368683087, −4.07861740434068918112222930064, −3.35194308597820693139423438798, −1.71082725750893476049855083364, −0.55972199505217490081359278376,
0.55972199505217490081359278376, 1.71082725750893476049855083364, 3.35194308597820693139423438798, 4.07861740434068918112222930064, 5.26469628943002324733368683087, 6.72294132753306925612716542557, 7.83487879245742495409687790477, 8.336417309746932619673485196296, 9.244829370886954283150748800787, 9.992323951065508470229482929495
