| L(s) = 1 | + 1.08e4·3-s + 2.62e5·4-s − 1.98e6·5-s − 5.09e7·7-s − 2.70e8·9-s + 2.83e9·12-s − 2.14e10·15-s + 6.87e10·16-s − 2.16e11·17-s − 6.24e10·19-s − 5.20e11·20-s − 5.51e11·21-s + 1.26e11·25-s − 7.11e12·27-s − 1.33e13·28-s + 2.89e13·29-s + 1.01e14·35-s − 7.09e13·36-s + 6.52e14·41-s + 5.37e14·45-s + 7.42e14·48-s + 9.70e14·49-s − 2.34e15·51-s + 5.73e15·53-s − 6.74e14·57-s − 8.66e15·59-s − 5.62e15·60-s + ⋯ |
| L(s) = 1 | + 0.549·3-s + 4-s − 1.01·5-s − 1.26·7-s − 0.698·9-s + 0.549·12-s − 0.558·15-s + 16-s − 1.82·17-s − 0.193·19-s − 1.01·20-s − 0.693·21-s + 0.0331·25-s − 0.932·27-s − 1.26·28-s + 1.99·29-s + 1.28·35-s − 0.698·36-s + 1.99·41-s + 0.709·45-s + 0.549·48-s + 0.596·49-s − 1.00·51-s + 1.73·53-s − 0.106·57-s − 59-s − 0.558·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(1.569698436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569698436\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + p^{9} T \) |
| good | 2 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 3 | \( 1 - 10810 T + p^{18} T^{2} \) |
| 5 | \( 1 + 1985254 T + p^{18} T^{2} \) |
| 7 | \( 1 + 50982910 T + p^{18} T^{2} \) |
| 11 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 13 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 17 | \( 1 + 216651752350 T + p^{18} T^{2} \) |
| 19 | \( 1 + 62437037542 T + p^{18} T^{2} \) |
| 23 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 29 | \( 1 - 28956785336138 T + p^{18} T^{2} \) |
| 31 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 37 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 41 | \( 1 - 652243002714578 T + p^{18} T^{2} \) |
| 43 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 47 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 53 | \( 1 - 5739806619558650 T + p^{18} T^{2} \) |
| 61 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 67 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 71 | \( 1 + 56563270329694462 T + p^{18} T^{2} \) |
| 73 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 79 | \( 1 - 186548867397995762 T + p^{18} T^{2} \) |
| 83 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 89 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
| 97 | \( ( 1 - p^{9} T )( 1 + p^{9} T ) \) |
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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
−11.53295753629708916888933910962, −10.56612086627466476731861376004, −9.129414633071559297767085000262, −8.088725262019185802361117462785, −6.93897317884380294862650413420, −6.09591468881936247563605981003, −4.17580313281605674370252671609, −3.06037902721396444993818262132, −2.36257247690574953730209512152, −0.52675109206253678704580272709,
0.52675109206253678704580272709, 2.36257247690574953730209512152, 3.06037902721396444993818262132, 4.17580313281605674370252671609, 6.09591468881936247563605981003, 6.93897317884380294862650413420, 8.088725262019185802361117462785, 9.129414633071559297767085000262, 10.56612086627466476731861376004, 11.53295753629708916888933910962
