| 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
| \(L(4)\) |
\(\approx\) |
\(7.684836205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.684836205\) |
| \(L(\frac{9}{2})\) |
|
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
−11.70732298936743943465864129384, −11.32524948116218632638724840604, −11.07867204852855979996120383714, −10.13185864159170129534113717234, −9.748479085169804506193050118935, −9.543510308704735337167534255877, −8.535071173562963934896859873437, −8.333920905363091558931401492429, −7.81884342489894090561881122715, −6.90693607574396367075596928978, −6.38590882806762728553430876062, −6.27469648784538065193368910422, −4.97812955831832845828533645247, −4.79363745722126651437355044008, −4.36997866007954201027625576512, −3.51414102551764017985321573721, −2.38206342602659234053097815318, −1.78817811195298377724684824501, −1.28436501564841689318684595454, −0.847780916498529710546076764424,
0.847780916498529710546076764424, 1.28436501564841689318684595454, 1.78817811195298377724684824501, 2.38206342602659234053097815318, 3.51414102551764017985321573721, 4.36997866007954201027625576512, 4.79363745722126651437355044008, 4.97812955831832845828533645247, 6.27469648784538065193368910422, 6.38590882806762728553430876062, 6.90693607574396367075596928978, 7.81884342489894090561881122715, 8.333920905363091558931401492429, 8.535071173562963934896859873437, 9.543510308704735337167534255877, 9.748479085169804506193050118935, 10.13185864159170129534113717234, 11.07867204852855979996120383714, 11.32524948116218632638724840604, 11.70732298936743943465864129384
