| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 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 | \( 1 - 3.34e6T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08623600787994055154306234310, −10.11533149863880843927081204906, −9.255795412835257517702880994226, −7.52667233958788907967708992039, −6.41874779761724056545097154792, −5.62051129125620481688211297597, −4.45980649645551528454389719145, −3.58408179240720247237084091263, −0.75675877771302812716467400367, 0,
0.75675877771302812716467400367, 3.58408179240720247237084091263, 4.45980649645551528454389719145, 5.62051129125620481688211297597, 6.41874779761724056545097154792, 7.52667233958788907967708992039, 9.255795412835257517702880994226, 10.11533149863880843927081204906, 11.08623600787994055154306234310
