| L(s) = 1 | + 9.06·2-s + 9·3-s + 50.2·4-s − 92.5·5-s + 81.6·6-s + 41.3·7-s + 165.·8-s + 81·9-s − 839.·10-s + 195.·11-s + 452.·12-s − 1.03e3·13-s + 374.·14-s − 833.·15-s − 106.·16-s − 856.·17-s + 734.·18-s − 969.·19-s − 4.65e3·20-s + 371.·21-s + 1.77e3·22-s − 2.38e3·23-s + 1.49e3·24-s + 5.44e3·25-s − 9.38e3·26-s + 729·27-s + 2.07e3·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.57·4-s − 1.65·5-s + 0.925·6-s + 0.318·7-s + 0.915·8-s + 0.333·9-s − 2.65·10-s + 0.487·11-s + 0.906·12-s − 1.69·13-s + 0.511·14-s − 0.955·15-s − 0.103·16-s − 0.718·17-s + 0.534·18-s − 0.615·19-s − 2.60·20-s + 0.184·21-s + 0.781·22-s − 0.941·23-s + 0.528·24-s + 1.74·25-s − 2.72·26-s + 0.192·27-s + 0.500·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 103 | \( 1 - 1.06e4T \) |
| good | 2 | \( 1 - 9.06T + 32T^{2} \) |
| 5 | \( 1 + 92.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 41.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 195.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 856.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 969.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.38e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.96e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.31e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.73e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.03e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.32e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.62e4T + 8.58e9T^{2} \) |
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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
−10.88283818776883672607796953308, −9.388115430516888458056236318377, −8.118339833634718796706640326453, −7.38741634059738452616715186322, −6.40866514959438185534897406362, −4.75044606054773496794593790577, −4.33292806722835412803320987477, −3.33645946386305205903844193217, −2.23191230148384624496273758658, 0,
2.23191230148384624496273758658, 3.33645946386305205903844193217, 4.33292806722835412803320987477, 4.75044606054773496794593790577, 6.40866514959438185534897406362, 7.38741634059738452616715186322, 8.118339833634718796706640326453, 9.388115430516888458056236318377, 10.88283818776883672607796953308
