| L(s) = 1 | + 8·2-s − 18·3-s + 48·4-s + 50·5-s − 144·6-s + 98·7-s + 256·8-s + 243·9-s + 400·10-s + 82·11-s − 864·12-s + 210·13-s + 784·14-s − 900·15-s + 1.28e3·16-s − 340·17-s + 1.94e3·18-s − 506·19-s + 2.40e3·20-s − 1.76e3·21-s + 656·22-s + 3.38e3·23-s − 4.60e3·24-s + 1.87e3·25-s + 1.68e3·26-s − 2.91e3·27-s + 4.70e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.204·11-s − 1.73·12-s + 0.344·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.285·17-s + 1.41·18-s − 0.321·19-s + 1.34·20-s − 0.872·21-s + 0.288·22-s + 1.33·23-s − 1.63·24-s + 3/5·25-s + 0.487·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.646140912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.646140912\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 82 T + 147694 T^{2} - 82 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 210 T + 577522 T^{2} - 210 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 20 p T + 51190 T^{2} + 20 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 506 T + 3431406 T^{2} + 506 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3388 T + 15037966 T^{2} - 3388 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4500 T + 714758 p T^{2} - 4500 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10158 T + 61747774 T^{2} - 10158 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13464 T + 166398838 T^{2} - 13464 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11648 T + 62072494 T^{2} - 11648 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3584 T + 226792550 T^{2} - 3584 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14344 T + 33982942 T^{2} - 14344 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 41406 T + 1263420394 T^{2} - 41406 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1244 T - 260923274 T^{2} + 1244 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 58668 T + 2538406462 T^{2} - 58668 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 56000 T + 3143341910 T^{2} - 56000 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 54134 T + 3952100590 T^{2} - 54134 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1014 T + 237048346 T^{2} - 1014 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 46696 T + 6358333598 T^{2} + 46696 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 175728 T + 15369952438 T^{2} + 175728 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 50308 T + 3886698598 T^{2} + 50308 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 170910 T + 20246699314 T^{2} + 170910 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.70746984445138666093898703793, −11.39878192682109009345535919287, −10.80839440398865832907022878959, −10.74161742825752625746978484999, −9.803125436001821333729315588055, −9.698986072092105022105990582161, −8.512086512088065964776086896070, −8.345684366567595819533278521209, −7.24153719135381584956530562204, −6.96308316658134641204642284116, −6.33240466828786493806625059322, −5.97846619280212780655953014403, −5.32993148942643270332000102216, −5.05715387814916642166196802778, −4.21943557468806148180001967016, −4.08874187492880468110075336971, −2.57322551934268417187464955829, −2.52420465329674368690991337141, −1.07885336370431753989492189197, −1.05108816185357158030445305591,
1.05108816185357158030445305591, 1.07885336370431753989492189197, 2.52420465329674368690991337141, 2.57322551934268417187464955829, 4.08874187492880468110075336971, 4.21943557468806148180001967016, 5.05715387814916642166196802778, 5.32993148942643270332000102216, 5.97846619280212780655953014403, 6.33240466828786493806625059322, 6.96308316658134641204642284116, 7.24153719135381584956530562204, 8.345684366567595819533278521209, 8.512086512088065964776086896070, 9.698986072092105022105990582161, 9.803125436001821333729315588055, 10.74161742825752625746978484999, 10.80839440398865832907022878959, 11.39878192682109009345535919287, 11.70746984445138666093898703793
