| 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
| \(L(2)\) |
\(\approx\) |
\(1.005769615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005769615\) |
| \(L(\frac{5}{2})\) |
|
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
−10.57008077507323082531613537551, −10.10655963713706728285976052855, −10.09763049182849821061936742125, −9.379049872101267325123945917934, −9.176855228990429572957950978137, −8.292144243457909312595558124102, −8.236397924584690690118901958643, −7.23690921335504237567878646511, −7.14452600246217062658953551468, −7.02088306134278847687574229716, −6.10780251559037002648656274849, −5.28383253281939190261231757165, −4.85088696188358942287210006021, −4.79712136615398120286490716957, −3.96462540800582875329562099558, −3.35778008061874873853911664770, −3.00826073633262746989049859498, −2.07461690951247887983695099257, −1.67905153712234932726601862466, −0.25284674047600444708128237707,
0.25284674047600444708128237707, 1.67905153712234932726601862466, 2.07461690951247887983695099257, 3.00826073633262746989049859498, 3.35778008061874873853911664770, 3.96462540800582875329562099558, 4.79712136615398120286490716957, 4.85088696188358942287210006021, 5.28383253281939190261231757165, 6.10780251559037002648656274849, 7.02088306134278847687574229716, 7.14452600246217062658953551468, 7.23690921335504237567878646511, 8.236397924584690690118901958643, 8.292144243457909312595558124102, 9.176855228990429572957950978137, 9.379049872101267325123945917934, 10.09763049182849821061936742125, 10.10655963713706728285976052855, 10.57008077507323082531613537551
