| L(s) = 1 | − 2.90·2-s − 90.5·3-s − 119.·4-s + 240.·5-s + 262.·6-s + 1.39e3·7-s + 718.·8-s + 6.00e3·9-s − 698.·10-s − 499.·11-s + 1.08e4·12-s − 4.03e3·14-s − 2.17e4·15-s + 1.32e4·16-s + 8.98e3·17-s − 1.74e4·18-s + 1.31e4·19-s − 2.87e4·20-s − 1.25e5·21-s + 1.44e3·22-s + 2.61e4·23-s − 6.49e4·24-s − 2.01e4·25-s − 3.45e5·27-s − 1.66e5·28-s + 9.40e4·29-s + 6.32e4·30-s + ⋯ |
| L(s) = 1 | − 0.256·2-s − 1.93·3-s − 0.934·4-s + 0.861·5-s + 0.496·6-s + 1.53·7-s + 0.495·8-s + 2.74·9-s − 0.220·10-s − 0.113·11-s + 1.80·12-s − 0.392·14-s − 1.66·15-s + 0.807·16-s + 0.443·17-s − 0.703·18-s + 0.438·19-s − 0.804·20-s − 2.96·21-s + 0.0290·22-s + 0.448·23-s − 0.959·24-s − 0.257·25-s − 3.37·27-s − 1.43·28-s + 0.715·29-s + 0.427·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.128749428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.128749428\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 2.90T + 128T^{2} \) |
| 3 | \( 1 + 90.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 240.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.39e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 499.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 8.98e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.61e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.40e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.38e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.36e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.34e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.19e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.69e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.48e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.50e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.29e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.34e6T + 8.07e13T^{2} \) |
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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
−11.45741264818736042973196410973, −10.40951119045918743168155239908, −9.878471679617092080668218419185, −8.413582881895969713964167457156, −7.18212200703356598608705610718, −5.78201744869466467580330101670, −5.14284691615251895848321688136, −4.38874738921313331570755641018, −1.56110642515577581195038853361, −0.76269369492339387347057712658,
0.76269369492339387347057712658, 1.56110642515577581195038853361, 4.38874738921313331570755641018, 5.14284691615251895848321688136, 5.78201744869466467580330101670, 7.18212200703356598608705610718, 8.413582881895969713964167457156, 9.878471679617092080668218419185, 10.40951119045918743168155239908, 11.45741264818736042973196410973
