| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 9-s − 2·10-s + 2·11-s + 2·12-s − 3·13-s − 15-s − 4·16-s + 4·17-s + 2·18-s + 3·19-s − 2·20-s + 4·22-s + 25-s − 6·26-s + 27-s + 6·29-s − 2·30-s + 9·31-s − 8·32-s + 2·33-s + 8·34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.832·13-s − 0.258·15-s − 16-s + 0.970·17-s + 0.471·18-s + 0.688·19-s − 0.447·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 1.11·29-s − 0.365·30-s + 1.61·31-s − 1.41·32-s + 0.348·33-s + 1.37·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.127504242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.127504242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.52427092547101, −12.03663885167938, −11.67307770702233, −11.55106552982213, −10.78477871001871, −10.09402952554731, −9.665685280832486, −9.573428020845382, −8.641857255835292, −8.334340197066937, −7.855027481099975, −7.356261066833246, −6.752022322566622, −6.469381662151677, −5.932951582569645, −5.301996087983037, −4.700158160511436, −4.622013238916660, −4.007379917413102, −3.252231523374924, −3.146869595007093, −2.680892565878576, −1.895628323087483, −1.247958891705755, −0.5149206271431993,
0.5149206271431993, 1.247958891705755, 1.895628323087483, 2.680892565878576, 3.146869595007093, 3.252231523374924, 4.007379917413102, 4.622013238916660, 4.700158160511436, 5.301996087983037, 5.932951582569645, 6.469381662151677, 6.752022322566622, 7.356261066833246, 7.855027481099975, 8.334340197066937, 8.641857255835292, 9.573428020845382, 9.665685280832486, 10.09402952554731, 10.78477871001871, 11.55106552982213, 11.67307770702233, 12.03663885167938, 12.52427092547101
