| L(s) = 1 | − 4·4-s + 12·11-s + 16·16-s − 88·19-s − 252·29-s − 488·31-s − 960·41-s − 48·44-s + 490·49-s − 1.06e3·59-s + 724·61-s − 64·64-s − 1.94e3·71-s + 352·76-s − 2.48e3·79-s + 1.94e3·89-s − 3.01e3·101-s − 1.39e3·109-s + 1.00e3·116-s − 2.55e3·121-s + 1.95e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.328·11-s + 1/4·16-s − 1.06·19-s − 1.61·29-s − 2.82·31-s − 3.65·41-s − 0.164·44-s + 10/7·49-s − 2.35·59-s + 1.51·61-s − 1/8·64-s − 3.24·71-s + 0.531·76-s − 3.54·79-s + 2.31·89-s − 2.96·101-s − 1.22·109-s + 0.806·116-s − 1.91·121-s + 1.41·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06532996212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06532996212\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 230 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3742 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 244 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 8890 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 480 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 148198 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 152354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 231190 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 534 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 362 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 p^{2} T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 972 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 557134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1244 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 986758 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 972 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1823230 T^{2} + p^{6} T^{4} \) |
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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
−11.28792771328761270956840009680, −10.48292180218817679293983086584, −10.07028053936825470840838467737, −9.319161056615281971435649588189, −9.234924487656440155652482455088, −8.551350564588298319884632469671, −8.453582595804842633662376481600, −7.61413251134611543398527635079, −7.17139628380740085726333108329, −6.89716269430816931745288989482, −6.12810605545769911628586362908, −5.54542915168246541381691448978, −5.36001002868703519990996740950, −4.50819282125306543715797136716, −4.05187394831556959170616047256, −3.53776505664875903962805814942, −2.94003264906923443691323586094, −1.76125149571473727126673028783, −1.65770923225021208606745541639, −0.079904918750340081979244923998,
0.079904918750340081979244923998, 1.65770923225021208606745541639, 1.76125149571473727126673028783, 2.94003264906923443691323586094, 3.53776505664875903962805814942, 4.05187394831556959170616047256, 4.50819282125306543715797136716, 5.36001002868703519990996740950, 5.54542915168246541381691448978, 6.12810605545769911628586362908, 6.89716269430816931745288989482, 7.17139628380740085726333108329, 7.61413251134611543398527635079, 8.453582595804842633662376481600, 8.551350564588298319884632469671, 9.234924487656440155652482455088, 9.319161056615281971435649588189, 10.07028053936825470840838467737, 10.48292180218817679293983086584, 11.28792771328761270956840009680
