| L(s) = 1 | − 32·2-s + 162·3-s + 768·4-s − 5.18e3·6-s + 8.44e3·7-s − 1.63e4·8-s + 1.96e4·9-s + 450·11-s + 1.24e5·12-s − 1.38e5·13-s − 2.70e5·14-s + 3.27e5·16-s + 502·17-s − 6.29e5·18-s − 4.20e5·19-s + 1.36e6·21-s − 1.44e4·22-s − 1.86e6·23-s − 2.65e6·24-s + 4.44e6·26-s + 2.12e6·27-s + 6.48e6·28-s − 4.46e6·29-s − 5.84e6·31-s − 6.29e6·32-s + 7.29e4·33-s − 1.60e4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 1.32·7-s − 1.41·8-s + 9-s + 0.00926·11-s + 1.73·12-s − 1.34·13-s − 1.87·14-s + 5/4·16-s + 0.00145·17-s − 1.41·18-s − 0.739·19-s + 1.53·21-s − 0.0131·22-s − 1.39·23-s − 1.63·24-s + 1.90·26-s + 0.769·27-s + 1.99·28-s − 1.17·29-s − 1.13·31-s − 1.06·32-s + 0.0107·33-s − 0.00206·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 1206 p T + 921670 p^{2} T^{2} - 1206 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 450 T + 4168121982 T^{2} - 450 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 138906 T + 13829392730 T^{2} + 138906 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 502 T + 95028249770 T^{2} - 502 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 420168 T + 671508432614 T^{2} + 420168 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1865600 T + 4212468672926 T^{2} + 1865600 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4460022 T + 19660976652634 T^{2} + 4460022 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5844964 T + 55285489825566 T^{2} + 5844964 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9886806 T + 249948771833338 T^{2} + 9886806 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 47450232 T + 1212976058101278 T^{2} + 47450232 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 33009192 T + 947310406571302 T^{2} - 33009192 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1329500 T + 1364698721107934 T^{2} - 1329500 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 40356378 T + 3800925743113762 T^{2} - 40356378 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 22129650 T + 2704744653842478 T^{2} - 22129650 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 47225380 T + 14282938948822782 T^{2} - 47225380 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 392422932 T + 87985401564178150 T^{2} - 392422932 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 456284700 T + 112070407281736462 T^{2} + 456284700 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 380453364 T + 107265087653542450 T^{2} - 380453364 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 981509004 T + 471212362128096542 T^{2} + 981509004 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 34382104 T + 373979829800387510 T^{2} - 34382104 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 866729124 T + 764948875576324662 T^{2} + 866729124 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1611886176 T + 2158244304986466178 T^{2} + 1611886176 p^{9} T^{3} + p^{18} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80435326115697515476695756782, −10.41760894235313746977150076276, −9.817217590462301740778287774876, −9.573944632553141602589292518842, −8.690336159164183288333982327198, −8.664293727996877308798443024293, −7.87262755220590880432686436280, −7.78950416202392435746628477541, −7.00220756476028521538507383547, −6.77607096101092118598094965491, −5.53661128051725158713247412353, −5.23422872576420440895679174504, −4.15370869755916197113006065292, −3.79873870202164071342306699878, −2.69795244594746595067132688632, −2.29440403215304050875524756346, −1.67359892149663121545305711829, −1.40417690460341176191345694631, 0, 0,
1.40417690460341176191345694631, 1.67359892149663121545305711829, 2.29440403215304050875524756346, 2.69795244594746595067132688632, 3.79873870202164071342306699878, 4.15370869755916197113006065292, 5.23422872576420440895679174504, 5.53661128051725158713247412353, 6.77607096101092118598094965491, 7.00220756476028521538507383547, 7.78950416202392435746628477541, 7.87262755220590880432686436280, 8.664293727996877308798443024293, 8.690336159164183288333982327198, 9.573944632553141602589292518842, 9.817217590462301740778287774876, 10.41760894235313746977150076276, 10.80435326115697515476695756782
