| L(s) = 1 | − 107.·2-s + 729·3-s + 3.46e3·4-s + 5.61e3·5-s − 7.87e4·6-s − 4.41e5·7-s + 5.09e5·8-s + 5.31e5·9-s − 6.06e5·10-s − 9.58e6·11-s + 2.52e6·12-s − 1.04e6·13-s + 4.76e7·14-s + 4.09e6·15-s − 8.34e7·16-s + 8.94e7·17-s − 5.73e7·18-s − 1.31e8·19-s + 1.94e7·20-s − 3.21e8·21-s + 1.03e9·22-s − 6.51e8·23-s + 3.71e8·24-s − 1.18e9·25-s + 1.12e8·26-s + 3.87e8·27-s − 1.53e9·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.577·3-s + 0.423·4-s + 0.160·5-s − 0.688·6-s − 1.41·7-s + 0.687·8-s + 0.333·9-s − 0.191·10-s − 1.63·11-s + 0.244·12-s − 0.0598·13-s + 1.69·14-s + 0.0927·15-s − 1.24·16-s + 0.898·17-s − 0.397·18-s − 0.642·19-s + 0.0680·20-s − 0.818·21-s + 1.94·22-s − 0.917·23-s + 0.397·24-s − 0.974·25-s + 0.0713·26-s + 0.192·27-s − 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.1397302836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1397302836\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 + 107.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 5.61e3T + 1.22e9T^{2} \) |
| 7 | \( 1 + 4.41e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 9.58e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.04e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 8.94e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.31e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.51e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.82e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.29e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.75e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.34e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.01e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 7.21e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.09e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.31e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.73e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.31e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.72e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.25e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.25e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.90e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 9.75e12T + 6.73e25T^{2} \) |
| 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
−9.908844586171569599448654761397, −9.605226420234069417888168417257, −8.307084570165148012039558296665, −7.75358516943864713659708161960, −6.61739113995526397062241857294, −5.33717455392007927838078871377, −3.81442638248276381287122386478, −2.72020780821669169749221874574, −1.71058762880153823804227731767, −0.17756855605938558619263484376,
0.17756855605938558619263484376, 1.71058762880153823804227731767, 2.72020780821669169749221874574, 3.81442638248276381287122386478, 5.33717455392007927838078871377, 6.61739113995526397062241857294, 7.75358516943864713659708161960, 8.307084570165148012039558296665, 9.605226420234069417888168417257, 9.908844586171569599448654761397
