| L(s) = 1 | + 94·4-s + 1.18e4·7-s + 1.79e5·13-s + 3.05e5·16-s + 1.01e6·19-s − 2.62e6·25-s + 1.11e6·28-s + 8.89e5·31-s − 3.80e6·37-s + 1.15e7·43-s − 4.72e7·49-s + 1.68e7·52-s + 2.19e8·61-s + 8.12e7·64-s + 3.90e8·67-s + 1.19e9·73-s + 9.50e7·76-s + 4.98e7·79-s + 2.12e9·91-s + 1.93e9·97-s − 2.47e8·100-s + 3.16e9·103-s + 1.97e9·109-s + 3.61e9·112-s − 1.92e9·121-s + 8.35e7·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.183·4-s + 1.86·7-s + 1.74·13-s + 1.16·16-s + 1.78·19-s − 1.34·25-s + 0.342·28-s + 0.172·31-s − 0.333·37-s + 0.516·43-s − 1.17·49-s + 0.320·52-s + 2.03·61-s + 0.605·64-s + 2.36·67-s + 4.93·73-s + 0.326·76-s + 0.143·79-s + 3.25·91-s + 2.22·97-s − 0.247·100-s + 2.76·103-s + 1.34·109-s + 2.17·112-s − 0.815·121-s + 0.0317·124-s + 3.32·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(9.195248382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.195248382\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 - 47 p T^{2} - 9267 p^{5} T^{4} - 47 p^{19} T^{6} + p^{36} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 525844 p T^{2} + 57151543278 p^{3} T^{4} + 525844 p^{19} T^{6} + p^{36} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 5926 T + 10902561 p T^{2} - 5926 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 1923379196 T^{2} - 521940566421894090 T^{4} + 1923379196 p^{18} T^{6} + p^{36} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 89842 T + 21633487611 T^{2} - 89842 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 257308360340 T^{2} + \)\(36\!\cdots\!42\)\( T^{4} + 257308360340 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 505714 T + 161285269911 T^{2} - 505714 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 6624714167852 T^{2} + \)\(17\!\cdots\!14\)\( T^{4} + 6624714167852 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 29953210676468 T^{2} + \)\(63\!\cdots\!62\)\( T^{4} + 29953210676468 p^{18} T^{6} + p^{36} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 444568 T + 42450828990414 T^{2} - 444568 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 1902578 T + 251865513861675 T^{2} + 1902578 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 937891901045156 T^{2} + \)\(41\!\cdots\!90\)\( T^{4} + 937891901045156 p^{18} T^{6} + p^{36} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 5792512 T + 105947558797398 T^{2} - 5792512 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 2009990891848460 T^{2} + \)\(34\!\cdots\!62\)\( T^{4} + 2009990891848460 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 3449821563424876 T^{2} + \)\(20\!\cdots\!06\)\( T^{4} - 3449821563424876 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 18643239188687804 T^{2} + \)\(20\!\cdots\!70\)\( T^{4} + 18643239188687804 p^{18} T^{6} + p^{36} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 109941286 T + 20898050467726995 T^{2} - 109941286 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 195280618 T + 50211374185977279 T^{2} - 195280618 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 29645249718547292 T^{2} + \)\(35\!\cdots\!82\)\( T^{4} + 29645249718547292 p^{18} T^{6} + p^{36} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 598974754 T + 187289069793870579 T^{2} - 598974754 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 24907162 T + 239310264744889143 T^{2} - 24907162 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 665416715104736780 T^{2} + \)\(17\!\cdots\!62\)\( T^{4} + 665416715104736780 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 73360853505641164 T^{2} + \)\(22\!\cdots\!86\)\( T^{4} - 73360853505641164 p^{18} T^{6} + p^{36} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 967867258 T + 1521250911302105931 T^{2} - 967867258 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96157770419446398770079561974, −10.84962267442712759189532283193, −9.967254151733503670671170761804, −9.925071924354711601506677777377, −9.686611147231587149534311203549, −8.991932270204372432569052962876, −8.610316448326135491177583777183, −8.296357636722416581208824487911, −7.953057803269935285454796786615, −7.62188148666645997054177371824, −7.55574876224531037726053753667, −6.53556542279625651433471751225, −6.48774108849599889429442354599, −5.91300102340919663309451511663, −5.33090618048782670491700998025, −5.07685432850775138028618310258, −4.88508625943645532402998914181, −3.87191404487213253796328459671, −3.56124743951851293315171143398, −3.47997285810073373554924224322, −2.33574633783176467565860962157, −2.01852375909286850783971130713, −1.35384635297869716369668883638, −1.03596924526436548316924677411, −0.62967216230516064780684076388,
0.62967216230516064780684076388, 1.03596924526436548316924677411, 1.35384635297869716369668883638, 2.01852375909286850783971130713, 2.33574633783176467565860962157, 3.47997285810073373554924224322, 3.56124743951851293315171143398, 3.87191404487213253796328459671, 4.88508625943645532402998914181, 5.07685432850775138028618310258, 5.33090618048782670491700998025, 5.91300102340919663309451511663, 6.48774108849599889429442354599, 6.53556542279625651433471751225, 7.55574876224531037726053753667, 7.62188148666645997054177371824, 7.953057803269935285454796786615, 8.296357636722416581208824487911, 8.610316448326135491177583777183, 8.991932270204372432569052962876, 9.686611147231587149534311203549, 9.925071924354711601506677777377, 9.967254151733503670671170761804, 10.84962267442712759189532283193, 10.96157770419446398770079561974
