| L(s) = 1 | − 9.32·2-s + 9·3-s + 54.8·4-s − 99.7·5-s − 83.8·6-s − 63.0·7-s − 213.·8-s + 81·9-s + 930.·10-s − 579.·11-s + 493.·12-s + 47.4·13-s + 588.·14-s − 897.·15-s + 231.·16-s − 1.03e3·17-s − 755.·18-s − 2.27e3·19-s − 5.47e3·20-s − 567.·21-s + 5.40e3·22-s − 1.82e3·23-s − 1.91e3·24-s + 6.83e3·25-s − 442.·26-s + 729·27-s − 3.46e3·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 0.577·3-s + 1.71·4-s − 1.78·5-s − 0.951·6-s − 0.486·7-s − 1.17·8-s + 0.333·9-s + 2.94·10-s − 1.44·11-s + 0.990·12-s + 0.0778·13-s + 0.801·14-s − 1.03·15-s + 0.226·16-s − 0.872·17-s − 0.549·18-s − 1.44·19-s − 3.06·20-s − 0.280·21-s + 2.38·22-s − 0.718·23-s − 0.680·24-s + 2.18·25-s − 0.128·26-s + 0.192·27-s − 0.834·28-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
| \(L(3)\) |
\(\approx\) |
\(0.002708658860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002708658860\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.32T + 32T^{2} \) |
| 5 | \( 1 + 99.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 63.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 579.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 47.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.60e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.07e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.75e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.22e4T + 8.58e9T^{2} \) |
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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
−10.78604858126055641337962089286, −9.796471885802534811929701172780, −8.691168850426036725495156785068, −8.130454641293051514409336030680, −7.53357044725106891447075889532, −6.58813421936749202918991621342, −4.53475342805086368929800650929, −3.29514870742538927263334246258, −2.03952384575309806339482689380, −0.03378165009622883519991213483,
0.03378165009622883519991213483, 2.03952384575309806339482689380, 3.29514870742538927263334246258, 4.53475342805086368929800650929, 6.58813421936749202918991621342, 7.53357044725106891447075889532, 8.130454641293051514409336030680, 8.691168850426036725495156785068, 9.796471885802534811929701172780, 10.78604858126055641337962089286
