| L(s) = 1 | + 3-s − 4-s − 3·7-s − 2·9-s − 12-s − 13-s + 16-s − 7·19-s − 3·21-s + 7·25-s − 5·27-s + 3·28-s − 2·31-s + 2·36-s − 15·37-s − 39-s − 3·43-s + 48-s − 49-s + 52-s − 7·57-s − 11·61-s + 6·63-s − 64-s + 2·67-s + 5·73-s + 7·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 1.13·7-s − 2/3·9-s − 0.288·12-s − 0.277·13-s + 1/4·16-s − 1.60·19-s − 0.654·21-s + 7/5·25-s − 0.962·27-s + 0.566·28-s − 0.359·31-s + 1/3·36-s − 2.46·37-s − 0.160·39-s − 0.457·43-s + 0.144·48-s − 1/7·49-s + 0.138·52-s − 0.927·57-s − 1.40·61-s + 0.755·63-s − 1/8·64-s + 0.244·67-s + 0.585·73-s + 0.808·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47988 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47988 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.715446943149254491487611907435, −9.347998494493462298578719244659, −8.757645720337519736140609742668, −8.537027051600554345439867466210, −7.994702407229854528657695059550, −7.16594219632939632508174737270, −6.68428296847112244131289931713, −6.23949501190234532802186961585, −5.45514312798553154769447617687, −4.91776017326610741476113514845, −4.10389670531896771129402693647, −3.39610474730844628053633375558, −2.93029036781430136858572719949, −1.95492167468984458072295126324, 0,
1.95492167468984458072295126324, 2.93029036781430136858572719949, 3.39610474730844628053633375558, 4.10389670531896771129402693647, 4.91776017326610741476113514845, 5.45514312798553154769447617687, 6.23949501190234532802186961585, 6.68428296847112244131289931713, 7.16594219632939632508174737270, 7.994702407229854528657695059550, 8.537027051600554345439867466210, 8.757645720337519736140609742668, 9.347998494493462298578719244659, 9.715446943149254491487611907435
