| L(s) = 1 | + 24.9·2-s − 243·3-s − 1.42e3·4-s + 1.13e4·5-s − 6.07e3·6-s − 4.61e4·7-s − 8.67e4·8-s + 5.90e4·9-s + 2.83e5·10-s − 1.58e5·11-s + 3.45e5·12-s − 2.30e6·13-s − 1.15e6·14-s − 2.75e6·15-s + 7.47e5·16-s − 3.64e6·17-s + 1.47e6·18-s + 8.79e6·19-s − 1.61e7·20-s + 1.12e7·21-s − 3.97e6·22-s − 4.28e6·23-s + 2.10e7·24-s + 7.99e7·25-s − 5.76e7·26-s − 1.43e7·27-s + 6.56e7·28-s + ⋯ |
| L(s) = 1 | + 0.552·2-s − 0.577·3-s − 0.695·4-s + 1.62·5-s − 0.318·6-s − 1.03·7-s − 0.935·8-s + 0.333·9-s + 0.896·10-s − 0.297·11-s + 0.401·12-s − 1.72·13-s − 0.572·14-s − 0.937·15-s + 0.178·16-s − 0.622·17-s + 0.184·18-s + 0.814·19-s − 1.12·20-s + 0.598·21-s − 0.164·22-s − 0.138·23-s + 0.540·24-s + 1.63·25-s − 0.951·26-s − 0.192·27-s + 0.720·28-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
| \(L(6)\) |
\(\approx\) |
\(1.258760812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258760812\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 24.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.13e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.61e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.58e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.30e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.64e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.79e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.28e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.38e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.47e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.64e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.41e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.12e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + 9.13e7T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.00e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 3.63e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 7.77e8T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.23e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.80e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.41e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.91e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.60e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.00e9T + 7.15e21T^{2} \) |
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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
−10.37528944435996449280845758305, −9.627095585034066345919934825710, −9.200027939215127865932137443856, −7.29170433613647564239088029553, −6.15538966514523447358643044954, −5.48434374987456647273803323069, −4.65424701906903844925668936543, −3.12823559383491889164432144229, −2.07011620390077984642649580058, −0.46085836384142906950712020296,
0.46085836384142906950712020296, 2.07011620390077984642649580058, 3.12823559383491889164432144229, 4.65424701906903844925668936543, 5.48434374987456647273803323069, 6.15538966514523447358643044954, 7.29170433613647564239088029553, 9.200027939215127865932137443856, 9.627095585034066345919934825710, 10.37528944435996449280845758305
