L(s) = 1 | + 16.2·2-s + 27·3-s + 136.·4-s − 30.6·5-s + 439.·6-s − 1.35e3·7-s + 137.·8-s + 729·9-s − 498.·10-s + 1.82e3·11-s + 3.68e3·12-s − 1.17e3·13-s − 2.20e4·14-s − 828.·15-s − 1.52e4·16-s + 1.31e4·17-s + 1.18e4·18-s − 1.85e4·19-s − 4.18e3·20-s − 3.66e4·21-s + 2.96e4·22-s − 1.45e4·23-s + 3.71e3·24-s − 7.71e4·25-s − 1.90e4·26-s + 1.96e4·27-s − 1.85e5·28-s + ⋯ |
L(s) = 1 | + 1.43·2-s + 0.577·3-s + 1.06·4-s − 0.109·5-s + 0.829·6-s − 1.49·7-s + 0.0948·8-s + 0.333·9-s − 0.157·10-s + 0.413·11-s + 0.615·12-s − 0.148·13-s − 2.14·14-s − 0.0633·15-s − 0.929·16-s + 0.650·17-s + 0.479·18-s − 0.621·19-s − 0.116·20-s − 0.863·21-s + 0.594·22-s − 0.249·23-s + 0.0547·24-s − 0.987·25-s − 0.212·26-s + 0.192·27-s − 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 2 | \( 1 - 16.2T + 128T^{2} \) |
| 5 | \( 1 + 30.6T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.35e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.17e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.31e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.83e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.60e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.07e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.30e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.80e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.41e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.71e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.77e6T + 8.07e13T^{2} \) |
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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
−11.19166647160924987294990017239, −9.794765061088995776046229229792, −9.041923290540610470214229499600, −7.45018063363387845024241021243, −6.41542881747670883192379756482, −5.50289762625475951641870914506, −3.92623280529556797033940535543, −3.44932486333415949416586782098, −2.17546885803636557412165301784, 0,
2.17546885803636557412165301784, 3.44932486333415949416586782098, 3.92623280529556797033940535543, 5.50289762625475951641870914506, 6.41542881747670883192379756482, 7.45018063363387845024241021243, 9.041923290540610470214229499600, 9.794765061088995776046229229792, 11.19166647160924987294990017239