| L(s) = 1 | − 4·5-s − 28·13-s − 36·17-s − 38·25-s + 28·29-s − 60·37-s + 28·41-s + 34·49-s − 132·53-s + 164·61-s + 112·65-s + 132·73-s + 144·85-s + 60·89-s − 28·97-s + 188·101-s + 36·109-s − 196·113-s + 226·121-s + 268·125-s + 127-s + 131-s + 137-s + 139-s − 112·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4/5·5-s − 2.15·13-s − 2.11·17-s − 1.51·25-s + 0.965·29-s − 1.62·37-s + 0.682·41-s + 0.693·49-s − 2.49·53-s + 2.68·61-s + 1.72·65-s + 1.80·73-s + 1.69·85-s + 0.674·89-s − 0.288·97-s + 1.86·101-s + 0.330·109-s − 1.73·113-s + 1.86·121-s + 2.14·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.772·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5382645181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5382645181\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 226 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 578 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 542 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 898 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2914 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4162 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4258 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8962 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6946 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12226 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 5822 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93419634596090318517451759103, −11.21486566675248596889244402401, −11.16143362658393236229735680907, −10.35984086282839865123959021943, −9.956134996014836185363136008592, −9.452334346649521832683529695246, −9.073453734694354445342273556588, −8.228250312457131434450787519998, −8.171293141549021848794077033530, −7.28511305399002838900004033409, −7.12225340203589621196389418329, −6.52582807122093813898948206203, −5.88119456797643154478480408709, −4.92507351507816178150991040788, −4.80556751714483081881464706638, −4.08004074321703854957714452243, −3.45728328986719825095057622209, −2.41654847157823056630460370510, −2.07836287361525681548224471035, −0.34264756385951236198587250276,
0.34264756385951236198587250276, 2.07836287361525681548224471035, 2.41654847157823056630460370510, 3.45728328986719825095057622209, 4.08004074321703854957714452243, 4.80556751714483081881464706638, 4.92507351507816178150991040788, 5.88119456797643154478480408709, 6.52582807122093813898948206203, 7.12225340203589621196389418329, 7.28511305399002838900004033409, 8.171293141549021848794077033530, 8.228250312457131434450787519998, 9.073453734694354445342273556588, 9.452334346649521832683529695246, 9.956134996014836185363136008592, 10.35984086282839865123959021943, 11.16143362658393236229735680907, 11.21486566675248596889244402401, 11.93419634596090318517451759103
