| L(s) = 1 | − 11·5-s + 5·11-s + 5·13-s − 100·17-s + 67·19-s − 76·23-s − 111·25-s − 275·29-s + 362·31-s − 5·37-s + 162·41-s + 721·43-s + 216·47-s − 495·53-s − 55·55-s − 173·59-s − 532·61-s − 55·65-s + 111·67-s − 1.60e3·71-s + 1.21e3·73-s + 1.46e3·79-s + 1.40e3·83-s + 1.10e3·85-s + 1.97e3·89-s − 737·95-s + 561·97-s + ⋯ |
| L(s) = 1 | − 0.983·5-s + 0.137·11-s + 0.106·13-s − 1.42·17-s + 0.808·19-s − 0.689·23-s − 0.887·25-s − 1.76·29-s + 2.09·31-s − 0.0222·37-s + 0.617·41-s + 2.55·43-s + 0.670·47-s − 1.28·53-s − 0.134·55-s − 0.381·59-s − 1.11·61-s − 0.104·65-s + 0.202·67-s − 2.67·71-s + 1.94·73-s + 2.07·79-s + 1.86·83-s + 1.40·85-s + 2.35·89-s − 0.795·95-s + 0.587·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.404960847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.404960847\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 11 T + 232 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 304 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 3194 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 100 T + 11554 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 67 T + 14406 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 76 T + 6478 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 275 T + 61846 T^{2} + 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 362 T + 73043 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 3606 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 162 T + 128770 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 721 T + 288540 T^{2} - 721 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 216 T + 156778 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 495 T + 355102 T^{2} + 495 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 173 T + 282706 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 532 T + 447518 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 111 T + 318532 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1600 T + 1262410 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1215 T + 1011556 T^{2} - 1215 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1460 T + 1333505 T^{2} - 1460 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1409 T + 1147696 T^{2} - 1409 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1974 T + 2298994 T^{2} - 1974 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 561 T + 1863448 T^{2} - 561 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.148211027078777109503226112633, −8.867252081169705071130376864489, −8.121070559850045972099330977573, −7.924696513180929740046411067689, −7.55011037609730848341777665087, −7.46994500439216283179085862081, −6.54595364319892785895343739848, −6.50948730595937040675612361792, −5.90077993846028645910477535601, −5.62448134511790096343918824379, −4.91179631766756853317712722677, −4.52761841912903846309456187618, −3.99424322950204631847547786200, −3.96608009883371703867824881958, −3.20728861047828855995582904470, −2.75490993615749635287169925587, −2.08159470427853830569704756904, −1.73600251241790021850459588096, −0.64056766126831727329059723407, −0.53007006709219317987861747342,
0.53007006709219317987861747342, 0.64056766126831727329059723407, 1.73600251241790021850459588096, 2.08159470427853830569704756904, 2.75490993615749635287169925587, 3.20728861047828855995582904470, 3.96608009883371703867824881958, 3.99424322950204631847547786200, 4.52761841912903846309456187618, 4.91179631766756853317712722677, 5.62448134511790096343918824379, 5.90077993846028645910477535601, 6.50948730595937040675612361792, 6.54595364319892785895343739848, 7.46994500439216283179085862081, 7.55011037609730848341777665087, 7.924696513180929740046411067689, 8.121070559850045972099330977573, 8.867252081169705071130376864489, 9.148211027078777109503226112633
