| L(s) = 1 | − 90·2-s + 4.37e3·4-s − 1.44e3·5-s + 1.60e4·7-s − 1.50e5·8-s + 1.29e5·10-s + 5.08e5·11-s − 6.08e5·13-s − 1.44e6·14-s + 4.52e6·16-s − 4.20e6·19-s − 6.30e6·20-s − 4.57e7·22-s + 7.93e6·23-s − 8.38e6·25-s + 5.47e7·26-s + 7.04e7·28-s + 5.45e7·29-s + 2.60e7·31-s − 1.54e8·32-s − 2.31e7·35-s + 1.03e8·37-s + 3.78e8·38-s + 2.17e8·40-s − 3.11e8·41-s + 7.19e7·43-s + 2.22e9·44-s + ⋯ |
| L(s) = 1 | − 2.81·2-s + 4.27·4-s − 0.460·5-s + 0.957·7-s − 4.60·8-s + 1.29·10-s + 3.15·11-s − 1.63·13-s − 2.69·14-s + 4.31·16-s − 1.69·19-s − 1.96·20-s − 8.87·22-s + 1.23·23-s − 0.858·25-s + 4.61·26-s + 4.09·28-s + 2.65·29-s + 0.908·31-s − 4.60·32-s − 0.441·35-s + 1.49·37-s + 4.78·38-s + 2.12·40-s − 2.68·41-s + 0.489·43-s + 13.4·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.9765059498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9765059498\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 45 p T + 931 p^{2} T^{2} + 45 p^{11} T^{3} + p^{20} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 288 p T + 418273 p^{2} T^{2} + 288 p^{11} T^{3} + p^{20} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2299 p T - 479400 p^{2} T^{2} - 2299 p^{11} T^{3} + p^{20} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 508320 T + 112067165401 T^{2} - 508320 p^{10} T^{3} + p^{20} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 608639 T + 232582940472 T^{2} + 608639 p^{10} T^{3} + p^{20} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4031682981698 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2104549 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 7938720 T + 62434269626449 T^{2} - 7938720 p^{10} T^{3} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54504000 T + 1410935905300201 T^{2} - 54504000 p^{10} T^{3} + p^{20} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26014126 T - 142893535436925 T^{2} - 26014126 p^{10} T^{3} + p^{20} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 51946607 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 7588800 p T + 27181553709121 p^{2} T^{2} + 7588800 p^{11} T^{3} + p^{20} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 71964982 T - 16432523679023925 T^{2} - 71964982 p^{10} T^{3} + p^{20} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 259431840 T + 75034092104425249 T^{2} - 259431840 p^{10} T^{3} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 338704040537029298 T^{2} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1050202080 T + 878758222912750201 T^{2} + 1050202080 p^{10} T^{3} + p^{20} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 356794201 T - 586040809795654200 T^{2} - 356794201 p^{10} T^{3} + p^{20} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 591548989 T - 1472907598164839328 T^{2} - 591548989 p^{10} T^{3} + p^{20} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3038145413862016798 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 413274143 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1080519949 T - 8300752722439884600 T^{2} - 1080519949 p^{10} T^{3} + p^{20} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9874929600 T + 48020786055524573449 T^{2} + 9874929600 p^{10} T^{3} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 54253153184643612002 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13694177593 T + \)\(11\!\cdots\!00\)\( T^{2} - 13694177593 p^{10} T^{3} + p^{20} T^{4} \) |
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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
−12.05036955864902513170631961079, −11.79943707377085911074778925357, −11.36021838505029603646755380854, −10.69673586597023844809180271556, −10.02270166087780518131208944036, −9.813874097081060707605894872023, −8.908364318827383708335490017689, −8.888314257868456587898744252212, −8.306993088853002603649786320443, −7.78757194292681523644402111857, −6.87860730094234627509546114981, −6.87182611405513187985766148725, −6.06081760784287795208785710528, −4.62839024644009743446023389381, −4.28076235512331519498313196104, −3.11742074346703770168628404943, −2.05619616095561783737736176847, −1.62211222790639205104109718772, −0.75787215051470353880194229932, −0.67072884107178212512668186717,
0.67072884107178212512668186717, 0.75787215051470353880194229932, 1.62211222790639205104109718772, 2.05619616095561783737736176847, 3.11742074346703770168628404943, 4.28076235512331519498313196104, 4.62839024644009743446023389381, 6.06081760784287795208785710528, 6.87182611405513187985766148725, 6.87860730094234627509546114981, 7.78757194292681523644402111857, 8.306993088853002603649786320443, 8.888314257868456587898744252212, 8.908364318827383708335490017689, 9.813874097081060707605894872023, 10.02270166087780518131208944036, 10.69673586597023844809180271556, 11.36021838505029603646755380854, 11.79943707377085911074778925357, 12.05036955864902513170631961079
