| L(s) = 1 | + 23.0·2-s + 243·3-s − 1.51e3·4-s + 32.6·5-s + 5.61e3·6-s − 8.08e4·7-s − 8.22e4·8-s + 5.90e4·9-s + 754.·10-s − 1.96e5·11-s − 3.68e5·12-s − 2.11e6·13-s − 1.86e6·14-s + 7.94e3·15-s + 1.20e6·16-s − 4.62e6·17-s + 1.36e6·18-s + 4.04e6·19-s − 4.94e4·20-s − 1.96e7·21-s − 4.54e6·22-s − 2.90e7·23-s − 1.99e7·24-s − 4.88e7·25-s − 4.88e7·26-s + 1.43e7·27-s + 1.22e8·28-s + ⋯ |
| L(s) = 1 | + 0.510·2-s + 0.577·3-s − 0.739·4-s + 0.00467·5-s + 0.294·6-s − 1.81·7-s − 0.887·8-s + 0.333·9-s + 0.00238·10-s − 0.368·11-s − 0.427·12-s − 1.58·13-s − 0.927·14-s + 0.00269·15-s + 0.286·16-s − 0.789·17-s + 0.170·18-s + 0.374·19-s − 0.00345·20-s − 1.04·21-s − 0.187·22-s − 0.942·23-s − 0.512·24-s − 0.999·25-s − 0.806·26-s + 0.192·27-s + 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4537642219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4537642219\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 23.0T + 2.04e3T^{2} \) |
| 5 | \( 1 - 32.6T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.08e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.96e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.11e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.62e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.04e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.90e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.75e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.81e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.05e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.04e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.82e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.35e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.72e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 7.46e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.03e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.41e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.86e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 6.54e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.44e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.77e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.19e10T + 7.15e21T^{2} \) |
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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
−10.25404785299841933434573703703, −9.629553292671776976505026619562, −8.908276710164619254922356086434, −7.56887390117487366974191227207, −6.49125533726119618575286192694, −5.35473758333202429819142041556, −4.15812717600535827624152507332, −3.24376365420593909954272852762, −2.34162627639989883383467787237, −0.25476641786243324156357562424,
0.25476641786243324156357562424, 2.34162627639989883383467787237, 3.24376365420593909954272852762, 4.15812717600535827624152507332, 5.35473758333202429819142041556, 6.49125533726119618575286192694, 7.56887390117487366974191227207, 8.908276710164619254922356086434, 9.629553292671776976505026619562, 10.25404785299841933434573703703
