| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.270251619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270251619\) |
| \(L(3)\) |
|
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
−19.48202824504214113353894664780, −18.58739947477216633629722604007, −16.42249185595686761177404294704, −15.40289263271359717839663668376, −13.68117599380015502447117668360, −12.67522958378178130451058744024, −10.15212070800515013248741148543, −8.826023848599939488989504935568, −6.71607215381950625341138839890, −3.35298978555788219141257198948,
3.35298978555788219141257198948, 6.71607215381950625341138839890, 8.826023848599939488989504935568, 10.15212070800515013248741148543, 12.67522958378178130451058744024, 13.68117599380015502447117668360, 15.40289263271359717839663668376, 16.42249185595686761177404294704, 18.58739947477216633629722604007, 19.48202824504214113353894664780
