| L(s) = 1 | − 32·2-s − 162·3-s + 768·4-s + 5.18e3·6-s − 3.16e3·7-s − 1.63e4·8-s + 1.96e4·9-s + 2.70e4·11-s − 1.24e5·12-s − 7.08e4·13-s + 1.01e5·14-s + 3.27e5·16-s − 1.44e5·17-s − 6.29e5·18-s + 1.15e5·19-s + 5.12e5·21-s − 8.66e5·22-s − 7.99e5·23-s + 2.65e6·24-s + 2.26e6·26-s − 2.12e6·27-s − 2.43e6·28-s − 2.08e5·29-s + 6.36e6·31-s − 6.29e6·32-s − 4.38e6·33-s + 4.63e6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.498·7-s − 1.41·8-s + 9-s + 0.557·11-s − 1.73·12-s − 0.688·13-s + 0.704·14-s + 5/4·16-s − 0.420·17-s − 1.41·18-s + 0.203·19-s + 0.575·21-s − 0.788·22-s − 0.595·23-s + 1.63·24-s + 0.973·26-s − 0.769·27-s − 0.747·28-s − 0.0546·29-s + 1.23·31-s − 1.06·32-s − 0.644·33-s + 0.594·34-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 | $D_{4}$ | \( 1 + 3166 T - 1852071 p T^{2} + 3166 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 27084 T + 4803103546 T^{2} - 27084 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 70858 T + 16308846387 T^{2} + 70858 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 144756 T + 214619001478 T^{2} + 144756 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 115690 T + 436265121183 T^{2} - 115690 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 799668 T + 2741824392082 T^{2} + 799668 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 208140 T + 8938526934238 T^{2} + 208140 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6360094 T + 57405852057951 T^{2} - 6360094 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1876036 T + 24941642932878 T^{2} + 1876036 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 38125824 T + 904099447138066 T^{2} - 38125824 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3776162 T + 682777161430647 T^{2} - 3776162 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11199924 T + 1294585861381378 T^{2} - 11199924 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 38780952 T + 6907493154005242 T^{2} - 38780952 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 286804020 T + 37407547383589978 T^{2} - 286804020 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 255752666 T + 35405174494625571 T^{2} + 255752666 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 208870006 T + 48155607060534303 T^{2} + 208870006 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 277057536 T + 105156674810622286 T^{2} + 277057536 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 60155132 T + 67672504711861782 T^{2} - 60155132 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 127287760 T + 230560963243036638 T^{2} - 127287760 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 596984628 T + 330798104814677002 T^{2} + 596984628 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 940116480 T + 562146108317908018 T^{2} - 940116480 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1317925426 T + 906740973602065203 T^{2} + 1317925426 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−10.72084476947275836962078064388, −10.66666530653370478428432025170, −9.979087594405284503781934571681, −9.598444534577307483119370754307, −9.127350034207976838821958991492, −8.597991869028760443529777816626, −7.76762530754042668138085773648, −7.48267316450309454955014130451, −6.68483232221556197914502070034, −6.54462855124113036125561547229, −5.78305907312235730302001337344, −5.37718389589092065676110937388, −4.34152289607575701726245817652, −3.95371776279156984197884995413, −2.73918520692992314118570804859, −2.39903445905845334191450592233, −1.28761801814845106503651301109, −1.04345280845314758557678611129, 0, 0,
1.04345280845314758557678611129, 1.28761801814845106503651301109, 2.39903445905845334191450592233, 2.73918520692992314118570804859, 3.95371776279156984197884995413, 4.34152289607575701726245817652, 5.37718389589092065676110937388, 5.78305907312235730302001337344, 6.54462855124113036125561547229, 6.68483232221556197914502070034, 7.48267316450309454955014130451, 7.76762530754042668138085773648, 8.597991869028760443529777816626, 9.127350034207976838821958991492, 9.598444534577307483119370754307, 9.979087594405284503781934571681, 10.66666530653370478428432025170, 10.72084476947275836962078064388
