| L(s) = 1 | + 4·5-s + 8·7-s − 92·11-s − 100·13-s − 92·17-s + 4·19-s + 8·23-s − 46·25-s + 84·29-s + 384·31-s + 32·35-s + 172·37-s + 300·41-s + 300·43-s − 16·47-s − 446·49-s − 12·53-s − 368·55-s + 644·59-s − 292·61-s − 400·65-s − 172·67-s + 408·71-s + 412·73-s − 736·77-s + 400·79-s − 948·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 0.431·7-s − 2.52·11-s − 2.13·13-s − 1.31·17-s + 0.0482·19-s + 0.0725·23-s − 0.367·25-s + 0.537·29-s + 2.22·31-s + 0.154·35-s + 0.764·37-s + 1.14·41-s + 1.06·43-s − 0.0496·47-s − 1.30·49-s − 0.0311·53-s − 0.902·55-s + 1.42·59-s − 0.612·61-s − 0.763·65-s − 0.313·67-s + 0.681·71-s + 0.660·73-s − 1.08·77-s + 0.569·79-s − 1.25·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.162221960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162221960\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 510 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 92 T + 430 p T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 100 T + 5166 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 92 T + 5030 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 11370 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 84 T + 45742 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 384 T + 84158 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 172 T + 99294 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 300 T + 157270 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 300 T + 180314 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 114782 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 288382 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 644 T + 462170 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 292 T + 240078 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 172 T + 278250 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 408 T + 672766 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 412 T + 690678 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 400 T + 963870 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 948 T + 1360138 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 572 T + 845846 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2204 T + 2633478 T^{2} - 2204 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.662792480496163355446543600710, −9.441459499161729544598997185917, −8.651910157887449987180556197540, −8.376341288831722219521912745059, −7.85479912013808595673996317229, −7.72141285147767170603437662495, −7.20166815985792174207873748113, −6.85663982471742886323434682254, −6.04895556470567799614040457354, −5.98136192580979019012121248397, −5.06944870754815578145752451427, −5.01924924228876027218602333715, −4.58891629868991337732050299903, −4.23099582514183572993488495344, −3.13682051332412651484221418709, −2.70864990701583260923350479167, −2.23537267017876954408340918840, −2.21298656889584361449449615550, −0.910587434348081480585361841018, −0.29964300976567435285467700142,
0.29964300976567435285467700142, 0.910587434348081480585361841018, 2.21298656889584361449449615550, 2.23537267017876954408340918840, 2.70864990701583260923350479167, 3.13682051332412651484221418709, 4.23099582514183572993488495344, 4.58891629868991337732050299903, 5.01924924228876027218602333715, 5.06944870754815578145752451427, 5.98136192580979019012121248397, 6.04895556470567799614040457354, 6.85663982471742886323434682254, 7.20166815985792174207873748113, 7.72141285147767170603437662495, 7.85479912013808595673996317229, 8.376341288831722219521912745059, 8.651910157887449987180556197540, 9.441459499161729544598997185917, 9.662792480496163355446543600710
