| L(s) = 1 | + 32·2-s − 162·3-s + 768·4-s + 1.25e3·5-s − 5.18e3·6-s − 4.80e3·7-s + 1.63e4·8-s + 1.96e4·9-s + 4.00e4·10-s − 5.24e4·11-s − 1.24e5·12-s + 1.26e5·13-s − 1.53e5·14-s − 2.02e5·15-s + 3.27e5·16-s − 3.11e5·17-s + 6.29e5·18-s − 1.00e5·19-s + 9.60e5·20-s + 7.77e5·21-s − 1.67e6·22-s − 6.94e5·23-s − 2.65e6·24-s + 1.17e6·25-s + 4.05e6·26-s − 2.12e6·27-s − 3.68e6·28-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 − 1.08·11-s − 1.73·12-s + 1.22·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.903·17-s + 1.41·18-s − 0.177·19-s + 1.34·20-s + 0.872·21-s − 1.52·22-s − 0.517·23-s − 1.63·24-s + 3/5·25-s + 1.73·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 52464 T + 4939458582 T^{2} + 52464 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 126644 T + 23699417694 T^{2} - 126644 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 311212 T + 162579917926 T^{2} + 311212 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 100736 T - 131936315674 T^{2} + 100736 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 694272 T + 1471646067726 T^{2} + 694272 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7077636 T + 40902352456926 T^{2} + 7077636 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8384776 T + 40929115156142 T^{2} - 8384776 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10977980 T + 289253492310254 T^{2} - 10977980 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 23401372 T + 717757394196118 T^{2} + 23401372 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14957720 T + 837158359177350 T^{2} + 14957720 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8820880 T + 1746202254433534 T^{2} + 8820880 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 92607548 T + 8491356576781742 T^{2} + 92607548 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 44092952 T + 10617130230130630 T^{2} - 44092952 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 102702484 T + 9050236657803902 T^{2} + 102702484 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 108617112 T + 16148784685765430 T^{2} + 108617112 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 212895160 T + 13865601806855038 T^{2} - 212895160 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 268013300 T + 41057427596024726 T^{2} + 268013300 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 121850288 T + 65241990195292638 T^{2} - 121850288 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 173668744 T + 381420042920248054 T^{2} - 173668744 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 967306492 T + 895201446436666934 T^{2} + 967306492 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 834076244 T + 1543789132984071174 T^{2} + 834076244 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−10.63254772363235935334947330887, −10.31210557532137723800290942489, −9.591211204934771847002092917775, −9.343191120033033724496959932233, −8.217826378245930105954144689886, −7.985793682863643921679276298706, −7.00845379155446381447359125305, −6.52338591250136441706720559001, −6.37174963995048339691539716026, −5.78283552517952148704219612089, −5.32655761553251035566550061996, −4.99144045440536649929593737639, −4.06792311318930299595831084296, −3.92323746943584680102644635221, −2.75557236628928613429556603231, −2.69095493889685906945638578483, −1.47810782468677941048554629413, −1.45339601679433535348582850110, 0, 0,
1.45339601679433535348582850110, 1.47810782468677941048554629413, 2.69095493889685906945638578483, 2.75557236628928613429556603231, 3.92323746943584680102644635221, 4.06792311318930299595831084296, 4.99144045440536649929593737639, 5.32655761553251035566550061996, 5.78283552517952148704219612089, 6.37174963995048339691539716026, 6.52338591250136441706720559001, 7.00845379155446381447359125305, 7.985793682863643921679276298706, 8.217826378245930105954144689886, 9.343191120033033724496959932233, 9.591211204934771847002092917775, 10.31210557532137723800290942489, 10.63254772363235935334947330887
