L(s) = 1 | + 3·3-s − 2·7-s + 9·9-s − 22·13-s − 26·19-s − 6·21-s + 25·25-s + 27·27-s + 46·31-s + 26·37-s − 66·39-s + 22·43-s − 45·49-s − 78·57-s + 74·61-s − 18·63-s − 122·67-s − 46·73-s + 75·75-s + 142·79-s + 81·81-s + 44·91-s + 138·93-s + 2·97-s − 194·103-s − 214·109-s + 78·111-s + ⋯ |
L(s) = 1 | + 3-s − 2/7·7-s + 9-s − 1.69·13-s − 1.36·19-s − 2/7·21-s + 25-s + 27-s + 1.48·31-s + 0.702·37-s − 1.69·39-s + 0.511·43-s − 0.918·49-s − 1.36·57-s + 1.21·61-s − 2/7·63-s − 1.82·67-s − 0.630·73-s + 75-s + 1.79·79-s + 81-s + 0.483·91-s + 1.48·93-s + 2/97·97-s − 1.88·103-s − 1.96·109-s + 0.702·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.387521866\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387521866\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 2 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 22 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 26 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 - 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 26 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 - 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 74 T + p^{2} T^{2} \) |
| 67 | \( 1 + 122 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 46 T + p^{2} T^{2} \) |
| 79 | \( 1 - 142 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 2 T + p^{2} T^{2} \) |
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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
−15.12826666204129484815952020603, −14.47896171700672442849080540854, −13.17330298920262800649218692802, −12.23683198774988137568285726835, −10.40934092403520371517869078998, −9.365713217825202064216917469403, −8.085326697589198755121990191009, −6.77972021518288507623009847471, −4.54496271099688646562080731130, −2.62671465500727783804393539657,
2.62671465500727783804393539657, 4.54496271099688646562080731130, 6.77972021518288507623009847471, 8.085326697589198755121990191009, 9.365713217825202064216917469403, 10.40934092403520371517869078998, 12.23683198774988137568285726835, 13.17330298920262800649218692802, 14.47896171700672442849080540854, 15.12826666204129484815952020603