| 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 − 4.19e4·11-s − 1.24e5·12-s − 6.78e4·13-s + 1.53e5·14-s + 2.02e5·15-s + 3.27e5·16-s + 9.66e4·17-s + 6.29e5·18-s + 6.64e5·19-s − 9.60e5·20-s − 7.77e5·21-s − 1.34e6·22-s + 2.46e5·23-s − 2.65e6·24-s + 1.17e6·25-s − 2.17e6·26-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 − 0.864·11-s − 1.73·12-s − 0.658·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.280·17-s + 1.41·18-s + 1.16·19-s − 1.34·20-s − 0.872·21-s − 1.22·22-s + 0.183·23-s − 1.63·24-s + 3/5·25-s − 0.931·26-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 | $D_{4}$ | \( 1 + 41964 T + 3753827206 T^{2} + 41964 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 67856 T + 20203912230 T^{2} + 67856 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 96636 T + 73512481318 T^{2} - 96636 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 664276 T + 549699193302 T^{2} - 664276 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 246120 T + 1783861547326 T^{2} - 246120 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4304772 T + 973884423446 p T^{2} + 4304772 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4366148 T + 15946517903118 T^{2} + 4366148 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3575804 T + 263118987200958 T^{2} + 3575804 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 44782812 T + 1011679258273558 T^{2} - 44782812 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 22063936 T + 554367583485510 T^{2} - 22063936 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7082088 T + 1056646930685470 T^{2} - 7082088 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 123069984 T + 10141111665880630 T^{2} + 123069984 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 151552896 T + 20421902589479782 T^{2} + 151552896 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 69217700 T + 15301428596969982 T^{2} + 69217700 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 348097064 T + 75132760109064918 T^{2} + 348097064 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 194772348 T - 4099897801452962 T^{2} + 194772348 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 297178240 T + 103170745301690526 T^{2} - 297178240 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 356059832 T + 221477083511340894 T^{2} + 356059832 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 706246128 T + 483800920770578902 T^{2} + 706246128 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1122685860 T + 988618562352096118 T^{2} + 1122685860 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 939835400 T + 1238911258054787934 T^{2} + 939835400 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−10.80465307445071281830962094050, −10.49660859597103751975366097368, −9.392012232077290531521691552396, −9.356971813100073855919707035577, −7.959495369609874278009619049437, −7.85168568019716790520047851613, −7.26558137371720777604112086992, −7.07455261744373851604291335635, −6.01744734453201526260371593973, −5.77248833390383538765327100738, −5.16501547317144568418292121864, −4.90596898782119865598977281716, −4.13991627672748038487423640312, −3.99181269703457233551188513995, −2.81948959602216840233386024682, −2.76143056136437405592185634132, −1.39178323086869168301088602811, −1.37068065121334199539502826370, 0, 0,
1.37068065121334199539502826370, 1.39178323086869168301088602811, 2.76143056136437405592185634132, 2.81948959602216840233386024682, 3.99181269703457233551188513995, 4.13991627672748038487423640312, 4.90596898782119865598977281716, 5.16501547317144568418292121864, 5.77248833390383538765327100738, 6.01744734453201526260371593973, 7.07455261744373851604291335635, 7.26558137371720777604112086992, 7.85168568019716790520047851613, 7.959495369609874278009619049437, 9.356971813100073855919707035577, 9.392012232077290531521691552396, 10.49660859597103751975366097368, 10.80465307445071281830962094050
