| L(s) = 1 | + 6·3-s + 10·5-s + 14·7-s + 27·9-s + 16·11-s − 76·13-s + 60·15-s − 124·17-s + 96·19-s + 84·21-s + 16·23-s + 75·25-s + 108·27-s + 188·29-s + 120·31-s + 96·33-s + 140·35-s − 132·37-s − 456·39-s + 100·41-s + 536·43-s + 270·45-s + 928·47-s + 147·49-s − 744·51-s + 884·53-s + 160·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 0.438·11-s − 1.62·13-s + 1.03·15-s − 1.76·17-s + 1.15·19-s + 0.872·21-s + 0.145·23-s + 3/5·25-s + 0.769·27-s + 1.20·29-s + 0.695·31-s + 0.506·33-s + 0.676·35-s − 0.586·37-s − 1.87·39-s + 0.380·41-s + 1.90·43-s + 0.894·45-s + 2.88·47-s + 3/7·49-s − 2.04·51-s + 2.29·53-s + 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.958401214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.958401214\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 16 T - 474 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 76 T + 5806 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13638 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 96 T + 13974 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16 T + 15150 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 188 T + 34286 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 120 T + 60590 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 70814 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 100 T + 89142 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 536 T + 200086 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 928 T + 408830 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 884 T + 460350 T^{2} - 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1688 T + 1302310 T^{2} - 1688 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 136 T + 540446 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 508 T + 13078 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 432 T + 602142 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 584 T + 1172390 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1404 T + 1802390 T^{2} + 1404 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1188 T + 2161254 T^{2} + 1188 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.072878722837527939586142587901, −8.964732374148290049718885098898, −8.390349714946514036406620496908, −8.170553418036341349572950894061, −7.43674559107440031126683104077, −7.36304784469423767044651107649, −6.83793746674199064758600944486, −6.62476794686954530232890162912, −5.89928017710127307204745421995, −5.42883137835542741888512647821, −5.07976470934800908326140153468, −4.59365488604901013001050674847, −4.10610478009000953014212877359, −3.91742405337645694849038101474, −2.84102200032032578035217941925, −2.68245585844091986719158402201, −2.20731290710861500314980733708, −1.94194266562167804595090075421, −0.928614307959104555198829645602, −0.75359996927019315822836225641,
0.75359996927019315822836225641, 0.928614307959104555198829645602, 1.94194266562167804595090075421, 2.20731290710861500314980733708, 2.68245585844091986719158402201, 2.84102200032032578035217941925, 3.91742405337645694849038101474, 4.10610478009000953014212877359, 4.59365488604901013001050674847, 5.07976470934800908326140153468, 5.42883137835542741888512647821, 5.89928017710127307204745421995, 6.62476794686954530232890162912, 6.83793746674199064758600944486, 7.36304784469423767044651107649, 7.43674559107440031126683104077, 8.170553418036341349572950894061, 8.390349714946514036406620496908, 8.964732374148290049718885098898, 9.072878722837527939586142587901
