| L(s) = 1 | − 36·3-s + 810·9-s + 312·13-s + 384·17-s − 2.42e3·23-s + 6.62e3·25-s − 1.45e4·27-s − 3.96e3·29-s − 1.12e4·39-s − 4.83e3·43-s + 4.91e4·49-s − 1.38e4·51-s − 1.82e3·53-s − 1.00e5·61-s + 8.72e4·69-s − 2.38e5·75-s + 2.05e5·79-s + 2.29e5·81-s + 1.42e5·87-s + 1.09e5·101-s + 8.31e4·103-s − 5.00e5·107-s − 3.44e5·113-s + 2.52e5·117-s + 6.17e5·121-s + 127-s + 1.73e5·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s + 0.512·13-s + 0.322·17-s − 0.955·23-s + 2.11·25-s − 3.84·27-s − 0.874·29-s − 1.18·39-s − 0.398·43-s + 2.92·49-s − 0.744·51-s − 0.0891·53-s − 3.47·61-s + 2.20·69-s − 4.89·75-s + 3.70·79-s + 35/9·81-s + 2.01·87-s + 1.06·101-s + 0.772·103-s − 4.22·107-s − 2.53·113-s + 1.70·117-s + 3.83·121-s + 5.50e−6·127-s + 0.920·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4823859960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4823859960\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{4} \) |
| 13 | $D_{4}$ | \( 1 - 24 p T - 146 p^{3} T^{2} - 24 p^{6} T^{3} + p^{10} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6624 T^{2} + 1083838 p^{2} T^{4} - 6624 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 49132 T^{2} + 1140403638 T^{4} - 49132 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 617088 T^{2} + 146890430542 T^{4} - 617088 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 192 T + 2791006 T^{2} - 192 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 4292764 T^{2} + 9023685324870 T^{4} - 4292764 p^{10} T^{6} + p^{20} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 1212 T + 3450766 T^{2} + 1212 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 1980 T + 13967182 T^{2} + 1980 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 72558796 T^{2} + 2754768158742 p^{2} T^{4} - 72558796 p^{10} T^{6} + p^{20} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 191019364 T^{2} + 17466588618543606 T^{4} - 191019364 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 259347072 T^{2} + 41835734180463454 T^{4} - 259347072 p^{10} T^{6} + p^{20} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2416 T + 284123046 T^{2} + 2416 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 407420832 T^{2} + 124761975513464638 T^{4} - 407420832 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 912 T + 836540998 T^{2} + 912 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 151308144 T^{2} + 442844399562129262 T^{4} - 151308144 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 50496 T + 2324330710 T^{2} + 50496 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 45114940 T^{2} - 516263867957300538 T^{4} - 45114940 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 6784475376 T^{2} + 18015098428504544062 T^{4} - 6784475376 p^{10} T^{6} + p^{20} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 4119671084 T^{2} + 12172896815453134278 T^{4} + 4119671084 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 102672 T + 7610525470 T^{2} - 102672 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 12846419184 T^{2} + 72288762441539676238 T^{4} - 12846419184 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 9461922432 T^{2} + 70529599078905653662 T^{4} + 9461922432 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 23763592180 T^{2} + \)\(28\!\cdots\!94\)\( T^{4} - 23763592180 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−6.73548468769692341503113403145, −6.58480493649862435851448345709, −6.29614761626982955008685865880, −6.06465064269600518032788290438, −5.81543646847883110422149216338, −5.71133189284422302858144532982, −5.53010982580691734201868019258, −5.14791063465802708158219883836, −4.89284365950255543310569194522, −4.70802418925461832163991844027, −4.58801467785745897756071883492, −4.13507870525019502984986974555, −4.02465113623678247740999029633, −3.58121956570458921271735987118, −3.43012528962717217230170215135, −3.12873385530515772411013333359, −2.55035684222680872200842907481, −2.48837013734943451908246258424, −1.92657821483756308793116593908, −1.65930055932560466870314747301, −1.42824408489393974091797167423, −0.987217426542848906393094559167, −0.70759572161363514706061269632, −0.62804971296690877803214317371, −0.11036701592707312095807058524,
0.11036701592707312095807058524, 0.62804971296690877803214317371, 0.70759572161363514706061269632, 0.987217426542848906393094559167, 1.42824408489393974091797167423, 1.65930055932560466870314747301, 1.92657821483756308793116593908, 2.48837013734943451908246258424, 2.55035684222680872200842907481, 3.12873385530515772411013333359, 3.43012528962717217230170215135, 3.58121956570458921271735987118, 4.02465113623678247740999029633, 4.13507870525019502984986974555, 4.58801467785745897756071883492, 4.70802418925461832163991844027, 4.89284365950255543310569194522, 5.14791063465802708158219883836, 5.53010982580691734201868019258, 5.71133189284422302858144532982, 5.81543646847883110422149216338, 6.06465064269600518032788290438, 6.29614761626982955008685865880, 6.58480493649862435851448345709, 6.73548468769692341503113403145
