| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 6·11-s − 2·15-s − 17-s + 2·19-s − 21-s − 25-s + 27-s + 4·29-s + 6·33-s + 2·35-s − 8·37-s + 2·41-s + 4·43-s − 2·45-s − 8·47-s + 49-s − 51-s + 14·53-s − 12·55-s + 2·57-s + 6·59-s + 10·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.516·15-s − 0.242·17-s + 0.458·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.04·33-s + 0.338·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 1.92·53-s − 1.61·55-s + 0.264·57-s + 0.781·59-s + 1.28·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.493228496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.493228496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.53896427000276, −14.83553681105035, −14.52771055620314, −13.88209126733963, −13.47633593436112, −12.72576027205494, −12.00726792811339, −11.90858479286709, −11.23428817291643, −10.55583922422763, −9.769416075026305, −9.455821922943372, −8.630641068672632, −8.469091615692384, −7.615561079366689, −6.945495229082729, −6.701715368141973, −5.856407411793535, −5.033826995321101, −4.137320961987834, −3.860408191679798, −3.275502846733683, −2.396350448950240, −1.480390359108618, −0.6490834347432710,
0.6490834347432710, 1.480390359108618, 2.396350448950240, 3.275502846733683, 3.860408191679798, 4.137320961987834, 5.033826995321101, 5.856407411793535, 6.701715368141973, 6.945495229082729, 7.615561079366689, 8.469091615692384, 8.630641068672632, 9.455821922943372, 9.769416075026305, 10.55583922422763, 11.23428817291643, 11.90858479286709, 12.00726792811339, 12.72576027205494, 13.47633593436112, 13.88209126733963, 14.52771055620314, 14.83553681105035, 15.53896427000276
