| L(s) = 1 | + 6·9-s + 32·11-s − 24·17-s + 96·19-s − 4·25-s + 88·41-s − 160·43-s + 68·49-s + 128·59-s − 128·67-s + 120·73-s + 27·81-s + 224·83-s + 312·89-s − 248·97-s + 192·99-s + 192·107-s + 328·113-s + 252·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 144·153-s + 157-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.90·11-s − 1.41·17-s + 5.05·19-s − 0.159·25-s + 2.14·41-s − 3.72·43-s + 1.38·49-s + 2.16·59-s − 1.91·67-s + 1.64·73-s + 1/3·81-s + 2.69·83-s + 3.50·89-s − 2.55·97-s + 1.93·99-s + 1.79·107-s + 2.90·113-s + 2.08·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 0.941·153-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.78391962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.78391962\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 486 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2886 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 16 T + 258 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 92 T^{2} - 17562 T^{4} - 92 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 12 T + 422 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 48 T + 1250 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 994 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3068 T^{2} + 3748518 T^{4} - 3068 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3292 T^{2} + 3990 p^{2} T^{4} - 3292 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 44 T + 3654 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 80 T + 4098 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2180 T^{2} + 8539014 T^{4} - 2180 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11068 T^{2} + 46399206 T^{4} - 11068 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 64 T + 2178 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7246 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 64 T + 7650 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 60 T + 8486 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 112 T + 16866 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 21158 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 124 T + 15750 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.15404520411360536959261807615, −7.03279678594536048608107320610, −6.72906601614634651144892364371, −6.62180958526508247118478975595, −6.30550312624459844750494765616, −6.10127489863562957711195252089, −5.87331250070446223472720876301, −5.50217590650576832476644276097, −5.27541617204073087150597512481, −4.98583084714534868305441122718, −4.96422704242433943734971048306, −4.44242384980541505104525153835, −4.37817944971091638444853891100, −3.93105496616566087931652034716, −3.61225773460644414728024091072, −3.53281312983569987776594881625, −3.41897154601584932408688757839, −2.92128031254901826818352893692, −2.71742430874883206792770510230, −2.01947445304148879221752937666, −1.87595515124117684380906450939, −1.56181215390899039200675499026, −1.02576225501406711924253252531, −0.804924344526649608344534354261, −0.69963351922745332010027911891,
0.69963351922745332010027911891, 0.804924344526649608344534354261, 1.02576225501406711924253252531, 1.56181215390899039200675499026, 1.87595515124117684380906450939, 2.01947445304148879221752937666, 2.71742430874883206792770510230, 2.92128031254901826818352893692, 3.41897154601584932408688757839, 3.53281312983569987776594881625, 3.61225773460644414728024091072, 3.93105496616566087931652034716, 4.37817944971091638444853891100, 4.44242384980541505104525153835, 4.96422704242433943734971048306, 4.98583084714534868305441122718, 5.27541617204073087150597512481, 5.50217590650576832476644276097, 5.87331250070446223472720876301, 6.10127489863562957711195252089, 6.30550312624459844750494765616, 6.62180958526508247118478975595, 6.72906601614634651144892364371, 7.03279678594536048608107320610, 7.15404520411360536959261807615
