| L(s) = 1 | + 2-s − 18·3-s − 5·4-s − 50·5-s − 18·6-s − 98·7-s + 21·8-s + 243·9-s − 50·10-s − 196·11-s + 90·12-s + 32·13-s − 98·14-s + 900·15-s − 941·16-s + 932·17-s + 243·18-s − 1.16e3·19-s + 250·20-s + 1.76e3·21-s − 196·22-s + 1.83e3·23-s − 378·24-s + 1.87e3·25-s + 32·26-s − 2.91e3·27-s + 490·28-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 1.15·3-s − 0.156·4-s − 0.894·5-s − 0.204·6-s − 0.755·7-s + 0.116·8-s + 9-s − 0.158·10-s − 0.488·11-s + 0.180·12-s + 0.0525·13-s − 0.133·14-s + 1.03·15-s − 0.918·16-s + 0.782·17-s + 0.176·18-s − 0.739·19-s + 0.139·20-s + 0.872·21-s − 0.0863·22-s + 0.722·23-s − 0.133·24-s + 3/5·25-s + 0.00928·26-s − 0.769·27-s + 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 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 | 2 | $D_{4}$ | \( 1 - T + 3 p T^{2} - p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 196 T + 308406 T^{2} + 196 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 32 T + 585334 T^{2} - 32 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 932 T + 536742 T^{2} - 932 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1164 T + 5021574 T^{2} + 1164 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1832 T + 12634350 T^{2} - 1832 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1108 T + 1368454 p T^{2} + 1108 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 13740 T + 95320670 T^{2} + 13740 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8244 T + 127461566 T^{2} + 8244 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2620 T + 231937302 T^{2} - 2620 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 15472 T + 166806454 T^{2} - 15472 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9016 T + 476864750 T^{2} + 9016 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 24752 T + 971549190 T^{2} - 24752 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 38224 T + 1287508934 T^{2} + 38224 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14868 T + 1269327086 T^{2} + 14868 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 65464 T + 2798174950 T^{2} - 65464 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 34244 T + 3748750286 T^{2} - 34244 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 59696 T + 5009152462 T^{2} + 59696 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 150648 T + 11827635102 T^{2} + 150648 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 82832 T + 9102131510 T^{2} + 82832 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 136284 T + 159850150 p T^{2} - 136284 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11064 T - 2593081554 T^{2} - 11064 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.62381768710417132051855740027, −12.01857459572127718924512715260, −11.51645954577320223194326985239, −10.96174249178507770327201953681, −10.62923189231887223385197968217, −10.10185367150574140531130932078, −9.081209062976673186189185161517, −9.058237015241066674602063854462, −7.943158414665710929452947643154, −7.33548482392018765304326688256, −6.96283141728926789465673463134, −6.25335912454967186543153112605, −5.45895171649117992811708795159, −5.08322767359306842987319934487, −4.10970989053742974069115959702, −3.71171079639272464169903205616, −2.61970301072063993751436939693, −1.35719866957792292636096446733, 0, 0,
1.35719866957792292636096446733, 2.61970301072063993751436939693, 3.71171079639272464169903205616, 4.10970989053742974069115959702, 5.08322767359306842987319934487, 5.45895171649117992811708795159, 6.25335912454967186543153112605, 6.96283141728926789465673463134, 7.33548482392018765304326688256, 7.943158414665710929452947643154, 9.058237015241066674602063854462, 9.081209062976673186189185161517, 10.10185367150574140531130932078, 10.62923189231887223385197968217, 10.96174249178507770327201953681, 11.51645954577320223194326985239, 12.01857459572127718924512715260, 12.62381768710417132051855740027
