L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 355.·5-s + 216·6-s + 343·7-s − 512·8-s + 729·9-s − 2.84e3·10-s + 1.72e3·11-s − 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 9.59e3·15-s + 4.09e3·16-s − 1.06e4·17-s − 5.83e3·18-s − 9.62e3·19-s + 2.27e4·20-s − 9.26e3·21-s − 1.37e4·22-s − 7.88e4·23-s + 1.38e4·24-s + 4.80e4·25-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 + 1.27·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.898·10-s + 0.390·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.733·15-s + 0.250·16-s − 0.524·17-s − 0.235·18-s − 0.321·19-s + 0.635·20-s − 0.218·21-s − 0.275·22-s − 1.35·23-s + 0.204·24-s + 0.615·25-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
\(L(4)\) |
\(\approx\) |
\(1.847124922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.847124922\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 - 355.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.72e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.06e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.62e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.88e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.19e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.45e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.74e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.04e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.70e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.87e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.69e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.79e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.86e6T + 8.07e13T^{2} \) |
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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
−9.866816133135385366539806709961, −8.854292811436844024017463334400, −8.058328748872874990935070984936, −6.73869933410005078024092327334, −6.21129432296773075157448063356, −5.30799832205463777867514463656, −4.12897588011408202744918128171, −2.48540638861174702096703143854, −1.66967850830902865010638128329, −0.69390456927513342232626589775,
0.69390456927513342232626589775, 1.66967850830902865010638128329, 2.48540638861174702096703143854, 4.12897588011408202744918128171, 5.30799832205463777867514463656, 6.21129432296773075157448063356, 6.73869933410005078024092327334, 8.058328748872874990935070984936, 8.854292811436844024017463334400, 9.866816133135385366539806709961