| L(s) = 1 | + 32·2-s + 162·3-s + 768·4-s − 248·5-s + 5.18e3·6-s − 1.30e4·7-s + 1.63e4·8-s + 1.96e4·9-s − 7.93e3·10-s − 8.31e4·11-s + 1.24e5·12-s − 5.71e4·13-s − 4.16e5·14-s − 4.01e4·15-s + 3.27e5·16-s − 3.87e5·17-s + 6.29e5·18-s − 1.52e6·19-s − 1.90e5·20-s − 2.10e6·21-s − 2.66e6·22-s − 3.18e5·23-s + 2.65e6·24-s − 2.25e6·25-s − 1.82e6·26-s + 2.12e6·27-s − 9.98e6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.177·5-s + 1.63·6-s − 2.04·7-s + 1.41·8-s + 9-s − 0.250·10-s − 1.71·11-s + 1.73·12-s − 0.554·13-s − 2.89·14-s − 0.204·15-s + 5/4·16-s − 1.12·17-s + 1.41·18-s − 2.67·19-s − 0.266·20-s − 2.36·21-s − 2.42·22-s − 0.237·23-s + 1.63·24-s − 1.15·25-s − 0.784·26-s + 0.769·27-s − 3.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 248 T + 2317826 T^{2} + 248 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 13000 T + 14776458 p T^{2} + 13000 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 83140 T + 4057113050 T^{2} + 83140 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 387084 T + 112723477990 T^{2} + 387084 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 80040 p T + 1151819095070 T^{2} + 80040 p^{10} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 318920 T - 590985276226 T^{2} + 318920 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4171588 T + 18506724007166 T^{2} - 4171588 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6668752 T + 54862628298918 T^{2} + 6668752 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7653708 T + 192749179285838 T^{2} - 7653708 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5349864 T + 382152080361994 T^{2} + 5349864 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9645848 T - 227987529999930 T^{2} - 9645848 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 45295036 T + 1570792389334658 T^{2} + 45295036 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 80817084 T + 7925105583689902 T^{2} - 80817084 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 169730636 T + 14532161851817210 T^{2} + 169730636 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 133384348 T + 26391135408815358 T^{2} + 133384348 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 27500800 T + 54577581693371502 T^{2} + 27500800 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 401208260 T + 116494629731126930 T^{2} - 401208260 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 21876508 T + 28645102017101430 T^{2} - 21876508 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 642214304 T + 316729805252179230 T^{2} + 642214304 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 935734356 T + 559615366203691498 T^{2} + 935734356 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 868756080 T + 842361451365109786 T^{2} + 868756080 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1084497388 T + 1799888563286494662 T^{2} - 1084497388 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−12.69601492386476920426829126530, −12.25955247323196826389532669458, −11.14991893778991966225194986658, −10.72115632770540666952275911102, −9.922699574711583926113137280397, −9.861804312203040618623972928855, −8.720853938532921322464017186036, −8.364103687574096774837399474453, −7.42863972168506397021807885289, −7.05868209040636341966794622584, −6.11837240698962232609559929167, −6.08481668117310702400345608987, −4.71805801584604914468627815124, −4.35386544639095445551038965597, −3.53251958413007608351893975171, −3.01330219522056322260123091978, −2.33837966827810690219597306398, −2.05363683632723099328499290677, 0, 0,
2.05363683632723099328499290677, 2.33837966827810690219597306398, 3.01330219522056322260123091978, 3.53251958413007608351893975171, 4.35386544639095445551038965597, 4.71805801584604914468627815124, 6.08481668117310702400345608987, 6.11837240698962232609559929167, 7.05868209040636341966794622584, 7.42863972168506397021807885289, 8.364103687574096774837399474453, 8.720853938532921322464017186036, 9.861804312203040618623972928855, 9.922699574711583926113137280397, 10.72115632770540666952275911102, 11.14991893778991966225194986658, 12.25955247323196826389532669458, 12.69601492386476920426829126530
