| L(s) = 1 | − 729·3-s + 4.09e3·4-s + 1.53e5·7-s + 5.31e5·9-s − 2.98e6·12-s + 9.39e6·13-s + 1.67e7·16-s + 1.78e7·19-s − 1.11e8·21-s − 3.87e8·27-s + 6.28e8·28-s − 5.30e8·31-s + 2.17e9·36-s − 2.82e9·37-s − 6.85e9·39-s + 2.35e8·43-s − 1.22e10·48-s + 9.72e9·49-s + 3.84e10·52-s − 1.30e10·57-s + 7.40e10·61-s + 8.15e10·63-s + 6.87e10·64-s + 1.51e11·67-s − 1.04e11·73-s + 7.32e10·76-s − 4.44e11·79-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 1.30·7-s + 9-s − 12-s + 1.94·13-s + 16-s + 0.380·19-s − 1.30·21-s − 27-s + 1.30·28-s − 0.597·31-s + 36-s − 1.10·37-s − 1.94·39-s + 0.0373·43-s − 48-s + 0.702·49-s + 1.94·52-s − 0.380·57-s + 1.43·61-s + 1.30·63-s + 64-s + 1.66·67-s − 0.690·73-s + 0.380·76-s − 1.82·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(2.832701704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832701704\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{6} T \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 7 | \( 1 - 153502 T + p^{12} T^{2} \) |
| 11 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 13 | \( 1 - 9397582 T + p^{12} T^{2} \) |
| 17 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 19 | \( 1 - 17886962 T + p^{12} T^{2} \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 31 | \( 1 + 530187838 T + p^{12} T^{2} \) |
| 37 | \( 1 + 2826257618 T + p^{12} T^{2} \) |
| 41 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 43 | \( 1 - 235885102 T + p^{12} T^{2} \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 61 | \( 1 - 74063873522 T + p^{12} T^{2} \) |
| 67 | \( 1 - 151031344462 T + p^{12} T^{2} \) |
| 71 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 73 | \( 1 + 104459767778 T + p^{12} T^{2} \) |
| 79 | \( 1 + 444304748158 T + p^{12} T^{2} \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( 1 - 1662757858942 T + p^{12} T^{2} \) |
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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
−11.61481695444534570459312059919, −11.21528381844666373910827217893, −10.34703734279190253931526763409, −8.494501547721573364448423868410, −7.32031862708110256630908441334, −6.18737690968055044414497721189, −5.23292862952214587992268179796, −3.73624489617432148742797693475, −1.80858070368851312242710260811, −1.02588038100240452358310312493,
1.02588038100240452358310312493, 1.80858070368851312242710260811, 3.73624489617432148742797693475, 5.23292862952214587992268179796, 6.18737690968055044414497721189, 7.32031862708110256630908441334, 8.494501547721573364448423868410, 10.34703734279190253931526763409, 11.21528381844666373910827217893, 11.61481695444534570459312059919
