| L(s) = 1 | − 33.1·2-s + 243·3-s − 950.·4-s − 7.61e3·5-s − 8.05e3·6-s + 3.18e4·7-s + 9.93e4·8-s + 5.90e4·9-s + 2.52e5·10-s − 9.91e4·11-s − 2.30e5·12-s − 2.33e6·13-s − 1.05e6·14-s − 1.85e6·15-s − 1.34e6·16-s + 6.72e6·17-s − 1.95e6·18-s + 1.10e7·19-s + 7.23e6·20-s + 7.74e6·21-s + 3.28e6·22-s − 5.57e7·23-s + 2.41e7·24-s + 9.16e6·25-s + 7.74e7·26-s + 1.43e7·27-s − 3.02e7·28-s + ⋯ |
| L(s) = 1 | − 0.732·2-s + 0.577·3-s − 0.463·4-s − 1.08·5-s − 0.422·6-s + 0.716·7-s + 1.07·8-s + 0.333·9-s + 0.797·10-s − 0.185·11-s − 0.267·12-s − 1.74·13-s − 0.524·14-s − 0.629·15-s − 0.320·16-s + 1.14·17-s − 0.244·18-s + 1.02·19-s + 0.505·20-s + 0.413·21-s + 0.135·22-s − 1.80·23-s + 0.618·24-s + 0.187·25-s + 1.27·26-s + 0.192·27-s − 0.332·28-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
| \(L(6)\) |
\(\approx\) |
\(0.8935225638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8935225638\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 33.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 7.61e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.18e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.91e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.33e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.72e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.10e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.57e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.29e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.79e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.52e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.17e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.54e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 5.43e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.96e8T + 9.26e18T^{2} \) |
| 61 | \( 1 - 2.08e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.67e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.37e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.74e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.30e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.24e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.87e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.05e9T + 7.15e21T^{2} \) |
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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
−10.15096309395775677874889616283, −9.823784581787111724705513837534, −8.277245768745755986328118232908, −8.017978139273954610174547000207, −7.18996498351941441917821545806, −5.11428414167419608831773902649, −4.35039100566052827458709528781, −3.13060753669128042880568287895, −1.68802234043928666228115252268, −0.47688717064910528209186432035,
0.47688717064910528209186432035, 1.68802234043928666228115252268, 3.13060753669128042880568287895, 4.35039100566052827458709528781, 5.11428414167419608831773902649, 7.18996498351941441917821545806, 8.017978139273954610174547000207, 8.277245768745755986328118232908, 9.823784581787111724705513837534, 10.15096309395775677874889616283
