| L(s) = 1 | − 2·3-s − 22·5-s + 6·9-s − 36·11-s − 42·13-s + 44·15-s − 8·17-s − 118·19-s + 104·23-s + 170·25-s − 70·27-s − 56·29-s + 20·31-s + 72·33-s + 504·37-s + 84·39-s + 544·41-s − 412·43-s − 132·45-s − 500·47-s + 16·51-s + 268·53-s + 792·55-s + 236·57-s − 198·59-s + 346·61-s + 924·65-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 1.96·5-s + 2/9·9-s − 0.986·11-s − 0.896·13-s + 0.757·15-s − 0.114·17-s − 1.42·19-s + 0.942·23-s + 1.35·25-s − 0.498·27-s − 0.358·29-s + 0.115·31-s + 0.379·33-s + 2.23·37-s + 0.344·39-s + 2.07·41-s − 1.46·43-s − 0.437·45-s − 1.55·47-s + 0.0439·51-s + 0.694·53-s + 1.94·55-s + 0.548·57-s − 0.436·59-s + 0.726·61-s + 1.76·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 22 T + 314 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 36 T + 934 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 42 T + 3410 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 7790 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 118 T + 7566 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 104 T + 26126 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 56 T + 21974 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 20 T + 48510 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 504 T + 159110 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 544 T + 193358 T^{2} - 544 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 412 T + 190278 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 500 T + 258974 T^{2} + 500 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 268 T + 20222 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 198 T + 410926 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 346 T + 429114 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1008 T + 591974 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1224 T + 980014 T^{2} - 1224 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 716 T + 832326 T^{2} + 716 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 584 T + 997470 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1230 T + 1395886 T^{2} + 1230 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 596 T + 39542 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 856 T + 1729230 T^{2} - 856 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.795338349435947992967751852845, −9.291176140765407696470979890003, −8.683354551130858191511040490215, −8.274982381961332666430683419545, −7.87181912634010456917747928539, −7.68468708630519855327284499705, −7.08252086145397983054296820161, −6.90750836444547593528123308403, −6.08805662436930298769336160273, −5.76676698803462285742128738095, −5.02788016898788949520669326312, −4.60446674060470503107048151255, −4.21915410579249301744866077101, −3.89954366024856406174705752654, −2.99982940197687107880046665667, −2.71908690192197231303328509031, −1.90106111339045793867328995115, −0.882086212581632631102598569213, 0, 0,
0.882086212581632631102598569213, 1.90106111339045793867328995115, 2.71908690192197231303328509031, 2.99982940197687107880046665667, 3.89954366024856406174705752654, 4.21915410579249301744866077101, 4.60446674060470503107048151255, 5.02788016898788949520669326312, 5.76676698803462285742128738095, 6.08805662436930298769336160273, 6.90750836444547593528123308403, 7.08252086145397983054296820161, 7.68468708630519855327284499705, 7.87181912634010456917747928539, 8.274982381961332666430683419545, 8.683354551130858191511040490215, 9.291176140765407696470979890003, 9.795338349435947992967751852845
