| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s + 3·5-s − 24·6-s − 14·7-s − 32·8-s + 27·9-s − 12·10-s + 35·11-s + 72·12-s + 26·13-s + 56·14-s + 18·15-s + 80·16-s − 45·17-s − 108·18-s − 63·19-s + 36·20-s − 84·21-s − 140·22-s + 79·23-s − 192·24-s − 75·25-s − 104·26-s + 108·27-s − 168·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.268·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.379·10-s + 0.959·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s + 0.309·15-s + 5/4·16-s − 0.642·17-s − 1.41·18-s − 0.760·19-s + 0.402·20-s − 0.872·21-s − 1.35·22-s + 0.716·23-s − 1.63·24-s − 3/5·25-s − 0.784·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.988105051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988105051\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 3 T + 84 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 35 T + 1454 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 45 T + 10164 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 63 T + 6466 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 79 T + 25726 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 161 T + 34900 T^{2} - 161 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 428 T + 102686 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 87684 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 530 T + 202010 T^{2} - 530 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 295 T + 142914 T^{2} + 295 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 532 T + 146494 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 104 T + 168550 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 680 T + 515590 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 531 T + 475828 T^{2} + 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 558 T + 565630 T^{2} - 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 676 T + 612014 T^{2} - 676 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 521 T + 373280 T^{2} - 521 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 262 T + 997182 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 514 T + 1208950 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1816 T + 2102494 T^{2} - 1816 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 684 T + 1770022 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.31741273188984311417772477057, −10.07604587570327575790661242359, −9.610246187058194191315401299209, −9.228123582903366872574349711660, −8.744725625392137467745407792799, −8.729856985870525128454084816797, −7.907953302906805534844621739498, −7.88076138773091518889227909007, −6.91691684016109867811734786092, −6.78462826264030740305049553691, −6.21186246423256542571891480884, −6.01936692331238868047561725943, −4.86653453925825502322449348609, −4.29736363505839285648186884479, −3.66356723028195560503314146163, −3.14719554725447090063504253799, −2.29541658489760747987674821940, −2.25242567477846948945551418043, −1.02812276399240854191916022978, −0.76321790974313645129960999364,
0.76321790974313645129960999364, 1.02812276399240854191916022978, 2.25242567477846948945551418043, 2.29541658489760747987674821940, 3.14719554725447090063504253799, 3.66356723028195560503314146163, 4.29736363505839285648186884479, 4.86653453925825502322449348609, 6.01936692331238868047561725943, 6.21186246423256542571891480884, 6.78462826264030740305049553691, 6.91691684016109867811734786092, 7.88076138773091518889227909007, 7.907953302906805534844621739498, 8.729856985870525128454084816797, 8.744725625392137467745407792799, 9.228123582903366872574349711660, 9.610246187058194191315401299209, 10.07604587570327575790661242359, 10.31741273188984311417772477057
