| L(s) = 1 | − 9·3-s + 94·7-s + 81·9-s + 146·13-s + 46·19-s − 846·21-s + 625·25-s − 729·27-s − 194·31-s − 2.06e3·37-s − 1.31e3·39-s + 3.21e3·43-s + 6.43e3·49-s − 414·57-s − 1.96e3·61-s + 7.61e3·63-s − 5.90e3·67-s − 8.54e3·73-s − 5.62e3·75-s − 7.68e3·79-s + 6.56e3·81-s + 1.37e4·91-s + 1.74e3·93-s − 1.88e4·97-s − 1.64e4·103-s + 2.20e4·109-s + 1.85e4·111-s + ⋯ |
| L(s) = 1 | − 3-s + 1.91·7-s + 9-s + 0.863·13-s + 0.127·19-s − 1.91·21-s + 25-s − 27-s − 0.201·31-s − 1.50·37-s − 0.863·39-s + 1.73·43-s + 2.68·49-s − 0.127·57-s − 0.528·61-s + 1.91·63-s − 1.31·67-s − 1.60·73-s − 75-s − 1.23·79-s + 81-s + 1.65·91-s + 0.201·93-s − 1.99·97-s − 1.54·103-s + 1.85·109-s + 1.50·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.370120488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370120488\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| good | 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 - 94 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 - 146 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 - 46 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 194 T + p^{4} T^{2} \) |
| 37 | \( 1 + 2062 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 3214 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 1966 T + p^{4} T^{2} \) |
| 67 | \( 1 + 5906 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 8542 T + p^{4} T^{2} \) |
| 79 | \( 1 + 7682 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 + 18814 T + p^{4} T^{2} \) |
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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
−14.95456193286921913286829754470, −13.80515904839706260006091379930, −12.32283566535191648311735339398, −11.27312125616230516058318833496, −10.60112652146086808925759879873, −8.678534999975324018407522027176, −7.32146882573243791034773119084, −5.64047629853814038035586552272, −4.46643341981148913093374685211, −1.37805448289011035027343985915,
1.37805448289011035027343985915, 4.46643341981148913093374685211, 5.64047629853814038035586552272, 7.32146882573243791034773119084, 8.678534999975324018407522027176, 10.60112652146086808925759879873, 11.27312125616230516058318833496, 12.32283566535191648311735339398, 13.80515904839706260006091379930, 14.95456193286921913286829754470
