| L(s) = 1 | + 243·3-s − 2.20e4·7-s + 5.90e4·9-s + 7.02e5·13-s + 2.90e6·19-s − 5.36e6·21-s + 9.76e6·25-s + 1.43e7·27-s − 4.93e7·31-s + 1.35e8·37-s + 1.70e8·39-s + 2.82e8·43-s + 2.05e8·49-s + 7.05e8·57-s − 1.97e8·61-s − 1.30e9·63-s + 1.43e9·67-s − 4.14e9·73-s + 2.37e9·75-s − 3.95e9·79-s + 3.48e9·81-s − 1.55e10·91-s − 1.19e10·93-s + 8.84e8·97-s + 3.33e9·103-s − 1.76e10·109-s + 3.28e10·111-s + ⋯ |
| L(s) = 1 | + 3-s − 1.31·7-s + 9-s + 1.89·13-s + 1.17·19-s − 1.31·21-s + 25-s + 27-s − 1.72·31-s + 1.94·37-s + 1.89·39-s + 1.92·43-s + 0.726·49-s + 1.17·57-s − 0.233·61-s − 1.31·63-s + 1.06·67-s − 1.99·73-s + 75-s − 1.28·79-s + 81-s − 2.48·91-s − 1.72·93-s + 0.103·97-s + 0.287·103-s − 1.14·109-s + 1.94·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.836812919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.836812919\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 7 | \( 1 + 22082 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( 1 - 702218 T + p^{10} T^{2} \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( 1 - 2901574 T + p^{10} T^{2} \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 + 49326674 T + p^{10} T^{2} \) |
| 37 | \( 1 - 135214586 T + p^{10} T^{2} \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( 1 - 282780982 T + p^{10} T^{2} \) |
| 47 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 197224726 T + p^{10} T^{2} \) |
| 67 | \( 1 - 1437442918 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 + 4144040686 T + p^{10} T^{2} \) |
| 79 | \( 1 + 3959005298 T + p^{10} T^{2} \) |
| 83 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 - 884916482 T + p^{10} T^{2} \) |
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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
−13.39018942387995697998367151567, −12.75162391711664467428948586589, −10.97301437570734955369130366334, −9.617805645742353414612173032460, −8.790106199198948945515747989949, −7.34869062361928273176351129853, −6.03171564719466672321865603560, −3.87837841852424688123569729195, −2.92294367546956325943393742701, −1.08433119731510685121959200391,
1.08433119731510685121959200391, 2.92294367546956325943393742701, 3.87837841852424688123569729195, 6.03171564719466672321865603560, 7.34869062361928273176351129853, 8.790106199198948945515747989949, 9.617805645742353414612173032460, 10.97301437570734955369130366334, 12.75162391711664467428948586589, 13.39018942387995697998367151567
