| L(s) = 1 | + 32·2-s + 243·3-s + 1.02e3·4-s + 3.63e3·5-s + 7.77e3·6-s + 3.29e4·7-s + 3.27e4·8-s + 5.90e4·9-s + 1.16e5·10-s − 7.58e5·11-s + 2.48e5·12-s − 2.48e6·13-s + 1.05e6·14-s + 8.82e5·15-s + 1.04e6·16-s + 8.29e6·17-s + 1.88e6·18-s − 1.08e7·19-s + 3.71e6·20-s + 8.00e6·21-s − 2.42e7·22-s + 2.05e7·23-s + 7.96e6·24-s − 3.56e7·25-s − 7.94e7·26-s + 1.43e7·27-s + 3.37e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.519·5-s + 0.408·6-s + 0.740·7-s + 0.353·8-s + 1/3·9-s + 0.367·10-s − 1.42·11-s + 0.288·12-s − 1.85·13-s + 0.523·14-s + 0.299·15-s + 1/4·16-s + 1.41·17-s + 0.235·18-s − 1.00·19-s + 0.259·20-s + 0.427·21-s − 1.00·22-s + 0.665·23-s + 0.204·24-s − 0.730·25-s − 1.31·26-s + 0.192·27-s + 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.682705027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.682705027\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 726 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 32936 T + p^{11} T^{2} \) |
| 11 | \( 1 + 758748 T + p^{11} T^{2} \) |
| 13 | \( 1 + 2482858 T + p^{11} T^{2} \) |
| 17 | \( 1 - 8290386 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10867300 T + p^{11} T^{2} \) |
| 23 | \( 1 - 20539272 T + p^{11} T^{2} \) |
| 29 | \( 1 - 28814550 T + p^{11} T^{2} \) |
| 31 | \( 1 - 150501392 T + p^{11} T^{2} \) |
| 37 | \( 1 + 8645722 p T + p^{11} T^{2} \) |
| 41 | \( 1 + 368008998 T + p^{11} T^{2} \) |
| 43 | \( 1 - 620469572 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2763110256 T + p^{11} T^{2} \) |
| 53 | \( 1 + 268284258 T + p^{11} T^{2} \) |
| 59 | \( 1 - 1672894740 T + p^{11} T^{2} \) |
| 61 | \( 1 + 7787197498 T + p^{11} T^{2} \) |
| 67 | \( 1 - 18706694156 T + p^{11} T^{2} \) |
| 71 | \( 1 + 8346990888 T + p^{11} T^{2} \) |
| 73 | \( 1 - 19641746522 T + p^{11} T^{2} \) |
| 79 | \( 1 + 5873807200 T + p^{11} T^{2} \) |
| 83 | \( 1 - 8492558172 T + p^{11} T^{2} \) |
| 89 | \( 1 - 75527864010 T + p^{11} T^{2} \) |
| 97 | \( 1 + 82356782494 T + p^{11} T^{2} \) |
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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
−20.82987495779085038416285715632, −19.10430707295029325127871182940, −17.23108626989743861177041588866, −15.20782876026591917702517667124, −14.03387651263070648020770498686, −12.45932901574772869611460080979, −10.20907494905427832351302466175, −7.72147821179438199604901250583, −5.08599830293239724140630579353, −2.43084861986843780768101640848,
2.43084861986843780768101640848, 5.08599830293239724140630579353, 7.72147821179438199604901250583, 10.20907494905427832351302466175, 12.45932901574772869611460080979, 14.03387651263070648020770498686, 15.20782876026591917702517667124, 17.23108626989743861177041588866, 19.10430707295029325127871182940, 20.82987495779085038416285715632
