| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s − 4·5-s + 24·6-s − 6·7-s + 32·8-s − 14·9-s − 16·10-s − 6·11-s + 72·12-s − 64·13-s − 24·14-s − 24·15-s + 80·16-s + 34·17-s − 56·18-s − 36·19-s − 48·20-s − 36·21-s − 24·22-s + 42·23-s + 192·24-s − 30·25-s − 256·26-s − 222·27-s − 72·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.357·5-s + 1.63·6-s − 0.323·7-s + 1.41·8-s − 0.518·9-s − 0.505·10-s − 0.164·11-s + 1.73·12-s − 1.36·13-s − 0.458·14-s − 0.413·15-s + 5/4·16-s + 0.485·17-s − 0.733·18-s − 0.434·19-s − 0.536·20-s − 0.374·21-s − 0.232·22-s + 0.380·23-s + 1.63·24-s − 0.239·25-s − 1.93·26-s − 1.58·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.699019655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.699019655\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 p T + 50 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 682 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T - 254 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 64 T + 2870 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 11494 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 42 T + 23722 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 428 T + 89374 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86 T + 54554 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 340 T + 122718 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 404 T + 77974 T^{2} - 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 620 T + 250902 T^{2} + 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 56 T - 76322 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 28 T - 2402 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 276 T + 418102 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 236 T + 437934 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 p T + 460358 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1542 T + 1290490 T^{2} - 1542 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 164 T + 468390 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1854 T + 1690954 T^{2} + 1854 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 372 T + 1121542 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1976 T + 2384782 T^{2} + 1976 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 220 T + 1754246 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−16.06589018941709298454780884351, −15.68595091456579959305492559192, −14.75344397286811820888050577933, −14.73762001750887540690387644714, −14.11406308092488643762483361989, −13.65430868128579697056332639616, −12.82358065527568854103235546054, −12.45028207467250388837514871948, −11.66528679248049696115233717724, −11.21281160793407976680008475735, −10.07519934686352208205065257099, −9.624582629398026452954370463022, −8.267629183879461858713189685157, −8.146306392884613605991514210834, −7.02168193591563058735953774380, −6.25861066872083709120680066208, −5.16522565065963445170565878562, −4.33657794288894325741571560855, −3.05819352511625889408467386416, −2.63634201067221819797768271326,
2.63634201067221819797768271326, 3.05819352511625889408467386416, 4.33657794288894325741571560855, 5.16522565065963445170565878562, 6.25861066872083709120680066208, 7.02168193591563058735953774380, 8.146306392884613605991514210834, 8.267629183879461858713189685157, 9.624582629398026452954370463022, 10.07519934686352208205065257099, 11.21281160793407976680008475735, 11.66528679248049696115233717724, 12.45028207467250388837514871948, 12.82358065527568854103235546054, 13.65430868128579697056332639616, 14.11406308092488643762483361989, 14.73762001750887540690387644714, 14.75344397286811820888050577933, 15.68595091456579959305492559192, 16.06589018941709298454780884351
