| L(s) = 1 | + 2.30e7·3-s + 1.07e9·4-s + 6.08e10·5-s + 3.24e14·9-s − 4.17e15·11-s + 2.47e16·12-s + 1.40e18·15-s + 1.15e18·16-s + 6.53e19·20-s − 2.09e20·23-s + 2.77e21·25-s + 2.72e21·27-s − 7.81e21·31-s − 9.61e22·33-s + 3.48e23·36-s − 5.98e23·37-s − 4.48e24·44-s + 1.97e25·45-s − 2.04e25·47-s + 2.65e25·48-s + 2.25e25·49-s − 1.44e26·53-s − 2.54e26·55-s + 7.08e26·59-s + 1.50e27·60-s + 1.23e27·64-s + 3.60e27·67-s + ⋯ |
| L(s) = 1 | + 1.60·3-s + 4-s + 1.99·5-s + 1.57·9-s − 11-s + 1.60·12-s + 3.20·15-s + 16-s + 1.99·20-s − 0.785·23-s + 2.98·25-s + 0.923·27-s − 0.333·31-s − 1.60·33-s + 1.57·36-s − 1.79·37-s − 44-s + 3.14·45-s − 1.69·47-s + 1.60·48-s + 49-s − 1.96·53-s − 1.99·55-s + 1.94·59-s + 3.20·60-s + 64-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{31}{2})\) |
\(\approx\) |
\(6.877523790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.877523790\) |
| \(L(16)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{15} T \) |
| good | 2 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 3 | \( 1 - 23027050 T + p^{30} T^{2} \) |
| 5 | \( 1 - 60892912874 T + p^{30} T^{2} \) |
| 7 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 13 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 17 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 19 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 23 | \( 1 + \)\(20\!\cdots\!50\)\( T + p^{30} T^{2} \) |
| 29 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 31 | \( 1 + \)\(78\!\cdots\!02\)\( T + p^{30} T^{2} \) |
| 37 | \( 1 + \)\(59\!\cdots\!50\)\( T + p^{30} T^{2} \) |
| 41 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 43 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 47 | \( 1 + \)\(20\!\cdots\!50\)\( T + p^{30} T^{2} \) |
| 53 | \( 1 + \)\(14\!\cdots\!50\)\( T + p^{30} T^{2} \) |
| 59 | \( 1 - \)\(70\!\cdots\!02\)\( T + p^{30} T^{2} \) |
| 61 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 67 | \( 1 - \)\(36\!\cdots\!50\)\( T + p^{30} T^{2} \) |
| 71 | \( 1 + \)\(71\!\cdots\!98\)\( T + p^{30} T^{2} \) |
| 73 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 79 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 83 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 89 | \( 1 - \)\(24\!\cdots\!98\)\( T + p^{30} T^{2} \) |
| 97 | \( 1 + \)\(12\!\cdots\!50\)\( T + p^{30} T^{2} \) |
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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
−13.80337385824507700098449042039, −12.82991099165546974511528344642, −10.47400305219808682724947411072, −9.616621020813711099974389358111, −8.263798994020487291501192883090, −6.82197616282946236038272902017, −5.45315477265960792309569738327, −3.13636192072397715695498051295, −2.24624725476629534424087021982, −1.63855387776507303539426990295,
1.63855387776507303539426990295, 2.24624725476629534424087021982, 3.13636192072397715695498051295, 5.45315477265960792309569738327, 6.82197616282946236038272902017, 8.263798994020487291501192883090, 9.616621020813711099974389358111, 10.47400305219808682724947411072, 12.82991099165546974511528344642, 13.80337385824507700098449042039
