L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s + 6·13-s + 14-s + 15-s + 16-s + 4·17-s + 18-s + 4·19-s − 20-s − 21-s + 23-s − 24-s + 25-s + 6·26-s − 27-s + 28-s − 8·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s − 1.48·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.857761409\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.857761409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.973557986881412818836476970791, −7.56599952206944281368783277013, −6.70527290395613656515884751813, −5.87004328384711505612906765128, −5.46088457442634571949672175545, −4.60078110471312864644789352836, −3.70380113236197646640472170556, −3.27900431891209090186420252012, −1.81500177403898795118038506805, −0.920161601076446811024041427299,
0.920161601076446811024041427299, 1.81500177403898795118038506805, 3.27900431891209090186420252012, 3.70380113236197646640472170556, 4.60078110471312864644789352836, 5.46088457442634571949672175545, 5.87004328384711505612906765128, 6.70527290395613656515884751813, 7.56599952206944281368783277013, 7.973557986881412818836476970791