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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.444242560\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444242560\) |
\(L(6)\) |
|
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
−17.75552076844496196671265457336, −16.50326712480877930152873954210, −15.05821331880388239608481344454, −13.29054855030938754713760335486, −11.59445282338386293578717628363, −10.65751015971245300546429465433, −8.332077640199756631035560704251, −6.26792424382981220068112714060, −4.53658006440230684502275490868, −1.24345938209537427108312650876,
1.24345938209537427108312650876, 4.53658006440230684502275490868, 6.26792424382981220068112714060, 8.332077640199756631035560704251, 10.65751015971245300546429465433, 11.59445282338386293578717628363, 13.29054855030938754713760335486, 15.05821331880388239608481344454, 16.50326712480877930152873954210, 17.75552076844496196671265457336