L(s) = 1 | + 2·2-s + 5·5-s + 34·7-s − 8·8-s + 10·10-s − 48·11-s + 70·13-s + 68·14-s − 16·16-s + 54·17-s + 238·19-s − 96·22-s − 51·23-s + 140·26-s − 30·29-s + 133·31-s + 108·34-s + 170·35-s + 436·37-s + 476·38-s − 40·40-s + 156·41-s + 88·43-s − 102·46-s − 516·47-s + 343·49-s + 1.27e3·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.447·5-s + 1.83·7-s − 0.353·8-s + 0.316·10-s − 1.31·11-s + 1.49·13-s + 1.29·14-s − 1/4·16-s + 0.770·17-s + 2.87·19-s − 0.930·22-s − 0.462·23-s + 1.05·26-s − 0.192·29-s + 0.770·31-s + 0.544·34-s + 0.821·35-s + 1.93·37-s + 2.03·38-s − 0.158·40-s + 0.594·41-s + 0.312·43-s − 0.326·46-s − 1.60·47-s + 49-s + 3.31·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(8.732796946\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.732796946\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 34 T + 813 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 89 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 27 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 119 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 51 T - 9566 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T - 23489 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 133 T - 12102 T^{2} - 133 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 218 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 156 T - 44585 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 88 T - 71763 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 516 T + 162433 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 639 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 654 T + 222337 T^{2} + 654 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 461 T - 14460 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 182 T - 267639 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 900 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 704 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1375 T + 1397586 T^{2} - 1375 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 915 T + 265438 T^{2} - 915 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1116 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T - 912417 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.04811115501800610198095171325, −9.754416193856721927091061615111, −9.192371727125854317855212162987, −8.805207847171216179722140953513, −8.192489835755638188117637805286, −7.82700186214416878592518621732, −7.67377028374569141440423726779, −7.28701456072090905241768753497, −6.28341548875111741944558127766, −5.92422470982393828001947677374, −5.66225936490475482568309067677, −5.01006049406086891840836102323, −4.90388750120373026849517716285, −4.37172201179398159031348965997, −3.46051263297646780713331201016, −3.32252180896008198088527012846, −2.50935458184100208720895349576, −1.91814730239620917499931530856, −1.04013047208876226146345326458, −0.898462269197590527335514611004,
0.898462269197590527335514611004, 1.04013047208876226146345326458, 1.91814730239620917499931530856, 2.50935458184100208720895349576, 3.32252180896008198088527012846, 3.46051263297646780713331201016, 4.37172201179398159031348965997, 4.90388750120373026849517716285, 5.01006049406086891840836102323, 5.66225936490475482568309067677, 5.92422470982393828001947677374, 6.28341548875111741944558127766, 7.28701456072090905241768753497, 7.67377028374569141440423726779, 7.82700186214416878592518621732, 8.192489835755638188117637805286, 8.805207847171216179722140953513, 9.192371727125854317855212162987, 9.754416193856721927091061615111, 10.04811115501800610198095171325