L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 6·11-s + 2·14-s + 16-s − 2·17-s − 19-s + 20-s + 6·22-s − 4·23-s + 25-s + 2·28-s + 8·29-s − 8·31-s + 32-s − 2·34-s + 2·35-s − 4·37-s − 38-s + 40-s + 4·41-s − 6·43-s + 6·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 1.80·11-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s + 0.223·20-s + 1.27·22-s − 0.834·23-s + 1/5·25-s + 0.377·28-s + 1.48·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s + 0.338·35-s − 0.657·37-s − 0.162·38-s + 0.158·40-s + 0.624·41-s − 0.914·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.476559357\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.476559357\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.224511827960553869968971159855, −8.644866374772568482009340887093, −7.62225325145996427252499523084, −6.67574875754100209128909614930, −6.17284293657032313546990620042, −5.16501990216446731794163022811, −4.32971829801054892278384141165, −3.60234198356452861464956029452, −2.23481316309274563835954274958, −1.35541568400212475297106193308,
1.35541568400212475297106193308, 2.23481316309274563835954274958, 3.60234198356452861464956029452, 4.32971829801054892278384141165, 5.16501990216446731794163022811, 6.17284293657032313546990620042, 6.67574875754100209128909614930, 7.62225325145996427252499523084, 8.644866374772568482009340887093, 9.224511827960553869968971159855