| L(s) = 1 | + 32·2-s − 243·3-s + 1.02e3·4-s − 1.33e4·5-s − 7.77e3·6-s + 1.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s − 4.26e5·10-s − 9.42e5·11-s − 2.48e5·12-s + 9.64e5·13-s + 5.37e5·14-s + 3.23e6·15-s + 1.04e6·16-s + 4.71e6·17-s + 1.88e6·18-s + 1.13e7·19-s − 1.36e7·20-s − 4.08e6·21-s − 3.01e7·22-s − 8.11e6·23-s − 7.96e6·24-s + 1.28e8·25-s + 3.08e7·26-s − 1.43e7·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.90·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.34·10-s − 1.76·11-s − 0.288·12-s + 0.720·13-s + 0.267·14-s + 1.10·15-s + 0.250·16-s + 0.805·17-s + 0.235·18-s + 1.05·19-s − 0.953·20-s − 0.218·21-s − 1.24·22-s − 0.262·23-s − 0.204·24-s + 2.64·25-s + 0.509·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.580454315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580454315\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 3 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 5 | \( 1 + 1.33e4T + 4.88e7T^{2} \) |
| 11 | \( 1 + 9.42e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 9.64e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.71e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.13e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 8.11e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.19e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.54e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.92e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.90e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.12e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.37e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.35e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 8.14e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.13e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.54e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.62e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.46e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.98e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.00e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.27e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.69e10T + 7.15e21T^{2} \) |
| 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
−13.40734289232772684124176962926, −12.21482080706868672842662423546, −11.45880397729062703342714201585, −10.51308975459225151848795165731, −8.083003077155108791338590858507, −7.40263381165934319913336191468, −5.52430661167136659613633580051, −4.36766268480360872213445505491, −3.09790766125587242941847261120, −0.71862013697632364180930231677,
0.71862013697632364180930231677, 3.09790766125587242941847261120, 4.36766268480360872213445505491, 5.52430661167136659613633580051, 7.40263381165934319913336191468, 8.083003077155108791338590858507, 10.51308975459225151848795165731, 11.45880397729062703342714201585, 12.21482080706868672842662423546, 13.40734289232772684124176962926
