| L(s) = 1 | + 2·2-s − 3·3-s + 5·5-s − 6·6-s − 8·8-s + 27·9-s + 10·10-s + 17·11-s + 162·13-s − 15·15-s − 16·16-s − 91·17-s + 54·18-s + 102·19-s + 34·22-s + 90·23-s + 24·24-s + 324·26-s − 216·27-s − 258·29-s − 30·30-s + 116·31-s − 51·33-s − 182·34-s − 314·37-s + 204·38-s − 486·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 9-s + 0.316·10-s + 0.465·11-s + 3.45·13-s − 0.258·15-s − 1/4·16-s − 1.29·17-s + 0.707·18-s + 1.23·19-s + 0.329·22-s + 0.815·23-s + 0.204·24-s + 2.44·26-s − 1.53·27-s − 1.65·29-s − 0.182·30-s + 0.672·31-s − 0.269·33-s − 0.918·34-s − 1.39·37-s + 0.870·38-s − 1.99·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.337603342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.337603342\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T - 2 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 17 T - 1042 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 81 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 91 T + 3368 T^{2} + 91 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 102 T + 3545 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 90 T - 4067 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 129 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 116 T - 16335 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 314 T + 47943 T^{2} + 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 124 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 434 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 497 T + 143186 T^{2} - 497 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 584 T + 192179 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 332 T - 95155 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 220 T - 178581 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 384 T - 153307 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 664 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 230 T - 336117 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 361 T - 362718 T^{2} + 361 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1172 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 40 T - 703369 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 175 T + p^{3} T^{2} )^{2} \) |
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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
−10.93868487050885149992253970110, −10.25172470764614335459675658527, −10.19416845241645532382559484007, −9.222801387746165772535965310548, −9.063776699326413119948421572735, −8.660128814831360795353086031558, −8.206855648214261860635940640943, −7.23226770220702600118584323188, −7.09415728351327891503901361864, −6.43694268484062779971202927091, −6.03363710362214331125014010561, −5.62060984453494494377280402562, −5.30438926951744579469088123154, −4.29091964297961246571130134263, −4.17172839891259934877407829716, −3.49835828537303443020942984406, −3.10406363662951226883615493286, −1.63477504343244096895413535310, −1.60366897607540790616288636030, −0.63938659695435979460917452141,
0.63938659695435979460917452141, 1.60366897607540790616288636030, 1.63477504343244096895413535310, 3.10406363662951226883615493286, 3.49835828537303443020942984406, 4.17172839891259934877407829716, 4.29091964297961246571130134263, 5.30438926951744579469088123154, 5.62060984453494494377280402562, 6.03363710362214331125014010561, 6.43694268484062779971202927091, 7.09415728351327891503901361864, 7.23226770220702600118584323188, 8.206855648214261860635940640943, 8.660128814831360795353086031558, 9.063776699326413119948421572735, 9.222801387746165772535965310548, 10.19416845241645532382559484007, 10.25172470764614335459675658527, 10.93868487050885149992253970110
