| L(s) = 1 | + 3.00e3·3-s + 6.55e4·4-s + 3.90e5·5-s + 5.76e6·7-s − 3.40e7·9-s + 1.45e8·11-s + 1.97e8·12-s + 1.57e9·13-s + 1.17e9·15-s + 4.29e9·16-s − 4.37e9·17-s + 2.56e10·20-s + 1.73e10·21-s + 1.52e11·25-s − 2.31e11·27-s + 3.77e11·28-s − 9.93e11·29-s + 4.36e11·33-s + 2.25e12·35-s − 2.22e12·36-s + 4.72e12·39-s + 9.50e12·44-s − 1.32e13·45-s − 4.28e13·47-s + 1.29e13·48-s + 3.32e13·49-s − 1.31e13·51-s + ⋯ |
| L(s) = 1 | + 0.458·3-s + 4-s + 5-s + 7-s − 0.789·9-s + 0.676·11-s + 0.458·12-s + 1.92·13-s + 0.458·15-s + 16-s − 0.626·17-s + 20-s + 0.458·21-s + 25-s − 0.820·27-s + 28-s − 1.98·29-s + 0.310·33-s + 35-s − 0.789·36-s + 0.882·39-s + 0.676·44-s − 0.789·45-s − 1.79·47-s + 0.458·48-s + 49-s − 0.287·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(4.688223755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.688223755\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{8} T \) |
| 7 | \( 1 - p^{8} T \) |
| good | 2 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 3 | \( 1 - 3007 T + p^{16} T^{2} \) |
| 11 | \( 1 - 145028447 T + p^{16} T^{2} \) |
| 13 | \( 1 - 1571306207 T + p^{16} T^{2} \) |
| 17 | \( 1 + 4372390753 T + p^{16} T^{2} \) |
| 19 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 23 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 29 | \( 1 + 993241792513 T + p^{16} T^{2} \) |
| 31 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 37 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 41 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 43 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 47 | \( 1 + 42819958052353 T + p^{16} T^{2} \) |
| 53 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 59 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 61 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 67 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 71 | \( 1 - 1283316773477762 T + p^{16} T^{2} \) |
| 73 | \( 1 + 491238137911678 T + p^{16} T^{2} \) |
| 79 | \( 1 - 1884802582005407 T + p^{16} T^{2} \) |
| 83 | \( 1 + 1568384056666558 T + p^{16} T^{2} \) |
| 89 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 97 | \( 1 - 11753087741306207 T + p^{16} T^{2} \) |
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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
−13.23360191880568210518298438740, −11.45328152828678743296113334979, −10.89427742545256314053215522265, −9.163398218271042231497877539516, −8.139179150757119012744377090900, −6.52112108955724933034356846376, −5.57351036169684802298973407338, −3.55831810863721276866193929483, −2.11872284066925610178040387314, −1.34855620339483840956975176640,
1.34855620339483840956975176640, 2.11872284066925610178040387314, 3.55831810863721276866193929483, 5.57351036169684802298973407338, 6.52112108955724933034356846376, 8.139179150757119012744377090900, 9.163398218271042231497877539516, 10.89427742545256314053215522265, 11.45328152828678743296113334979, 13.23360191880568210518298438740
