| L(s) = 1 | + 8·3-s + 250·5-s − 1.91e3·7-s + 1.61e3·9-s − 1.04e4·11-s − 5.90e3·13-s + 2.00e3·15-s + 6.40e3·17-s + 3.31e4·19-s − 1.52e4·21-s − 1.84e4·23-s + 4.68e4·25-s + 4.28e4·27-s + 5.36e4·29-s − 5.15e5·31-s − 8.39e4·33-s − 4.78e5·35-s + 4.68e5·37-s − 4.72e4·39-s − 1.40e5·41-s + 6.38e5·43-s + 4.04e5·45-s + 1.32e6·47-s + 1.24e6·49-s + 5.12e4·51-s − 1.93e6·53-s − 2.62e6·55-s + ⋯ |
| L(s) = 1 | + 0.171·3-s + 0.894·5-s − 2.10·7-s + 0.739·9-s − 2.37·11-s − 0.745·13-s + 0.153·15-s + 0.316·17-s + 1.10·19-s − 0.360·21-s − 0.316·23-s + 3/5·25-s + 0.419·27-s + 0.408·29-s − 3.10·31-s − 0.406·33-s − 1.88·35-s + 1.52·37-s − 0.127·39-s − 0.318·41-s + 1.22·43-s + 0.661·45-s + 1.86·47-s + 1.50·49-s + 0.0540·51-s − 1.78·53-s − 2.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T - 518 p T^{2} - 8 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1912 T + 2412422 T^{2} + 1912 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 10496 T + 59644582 T^{2} + 10496 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5908 T + 133842734 T^{2} + 5908 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6404 T + 127540966 T^{2} - 6404 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 33168 T + 1988210998 T^{2} - 33168 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 18440 T + 1290412390 T^{2} + 18440 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 53628 T + 19696248750 T^{2} - 53628 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 515664 T + 119022042142 T^{2} + 515664 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 468924 T + 242278266526 T^{2} - 468924 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 140700 T + 373449196662 T^{2} + 140700 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 638344 T + 214334453982 T^{2} - 638344 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1329000 T + 1185345421110 T^{2} - 1329000 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1939364 T + 3090850985534 T^{2} + 1939364 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1902320 T + 3499650737638 T^{2} + 1902320 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 400788 T + 3519445458382 T^{2} + 400788 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3427944 T + 14459122737134 T^{2} + 3427944 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1934768 T + 17411891922382 T^{2} + 1934768 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6458900 T + 30917638062294 T^{2} - 6458900 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2584288 T + 39242500979678 T^{2} + 2584288 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2032328 T + 49052200691534 T^{2} - 2032328 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8775860 T + 83406481716982 T^{2} - 8775860 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 19960380 T + 228121450352326 T^{2} + 19960380 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.98994409490204915280080166627, −10.67773418603842338012734227919, −10.26313509214395068042002898391, −9.683357686760193314239174400242, −9.370884929009476512175543081984, −9.222633075730619637379391622160, −7.960287097164014239857853447742, −7.63253349104631708010313153900, −7.15765224887291043723416429483, −6.55640948540121304203884775833, −5.69255838642876423125022588736, −5.62374607877356802281168624310, −4.85965498368969934105347809406, −3.96798266758593521770060380552, −3.06024376095262294604980101962, −2.83780337280396550081976261587, −2.18401963527299107172319233405, −1.19067708821943345224363080500, 0, 0,
1.19067708821943345224363080500, 2.18401963527299107172319233405, 2.83780337280396550081976261587, 3.06024376095262294604980101962, 3.96798266758593521770060380552, 4.85965498368969934105347809406, 5.62374607877356802281168624310, 5.69255838642876423125022588736, 6.55640948540121304203884775833, 7.15765224887291043723416429483, 7.63253349104631708010313153900, 7.960287097164014239857853447742, 9.222633075730619637379391622160, 9.370884929009476512175543081984, 9.683357686760193314239174400242, 10.26313509214395068042002898391, 10.67773418603842338012734227919, 10.98994409490204915280080166627
