L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 496.·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s + 3.97e3·10-s − 3.82e3·11-s − 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 1.34e4·15-s + 4.09e3·16-s − 2.14e3·17-s + 5.83e3·18-s − 1.85e4·19-s + 3.18e4·20-s + 9.26e3·21-s − 3.05e4·22-s − 6.75e4·23-s − 1.38e4·24-s + 1.68e5·25-s + 1.75e4·26-s − 1.96e4·27-s − 2.19e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.77·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.25·10-s − 0.866·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 1.02·15-s + 0.250·16-s − 0.106·17-s + 0.235·18-s − 0.620·19-s + 0.888·20-s + 0.218·21-s − 0.612·22-s − 1.15·23-s − 0.204·24-s + 2.16·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(4.152898077\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.152898077\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 5 | \( 1 - 496.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.82e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.14e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.75e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.57e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.81e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.32e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.14e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.52e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.34e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.47e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.20e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.61e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.80e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.11e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.00e6T + 8.07e13T^{2} \) |
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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
−10.03172431627339295695890855243, −8.964596857522459995945018859223, −7.68017054314935014675435478106, −6.36690088782038673274210091440, −6.08146777249720056357861997276, −5.23994311264526848513935228969, −4.26252800408652070030667469964, −2.73635179864832842677227804766, −2.05228556972014238253990509680, −0.810002406834040888644411851669,
0.810002406834040888644411851669, 2.05228556972014238253990509680, 2.73635179864832842677227804766, 4.26252800408652070030667469964, 5.23994311264526848513935228969, 6.08146777249720056357861997276, 6.36690088782038673274210091440, 7.68017054314935014675435478106, 8.964596857522459995945018859223, 10.03172431627339295695890855243