| L(s) = 1 | − 4·5-s − 2·7-s + 4·11-s + 2·17-s + 10·23-s + 10·25-s + 4·29-s − 14·31-s + 8·35-s + 14·37-s − 4·41-s − 22·43-s − 7·49-s + 8·53-s − 16·55-s + 4·61-s − 24·67-s + 28·71-s + 2·73-s − 8·77-s − 16·79-s + 6·83-s − 8·85-s + 8·89-s − 10·97-s + 8·101-s + 4·103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.755·7-s + 1.20·11-s + 0.485·17-s + 2.08·23-s + 2·25-s + 0.742·29-s − 2.51·31-s + 1.35·35-s + 2.30·37-s − 0.624·41-s − 3.35·43-s − 49-s + 1.09·53-s − 2.15·55-s + 0.512·61-s − 2.93·67-s + 3.32·71-s + 0.234·73-s − 0.911·77-s − 1.80·79-s + 0.658·83-s − 0.867·85-s + 0.847·89-s − 1.01·97-s + 0.796·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.765673289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765673289\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 11 T^{2} + 38 T^{4} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 9 T^{2} + 12 T^{3} - 152 T^{4} + 12 p T^{5} + 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + p T^{2} + 36 T^{3} + 216 T^{4} + 36 p T^{5} + p^{3} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 30 T^{2} + 18 T^{3} + 370 T^{4} + 18 p T^{5} + 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 26 T^{2} + 699 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 117 T^{2} - 30 p T^{3} + 4312 T^{4} - 30 p^{2} T^{5} + 117 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 69 T^{2} - 444 T^{3} + 2272 T^{4} - 444 p T^{5} + 69 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 14 T + 185 T^{2} + 1386 T^{3} + 9572 T^{4} + 1386 p T^{5} + 185 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 14 T + 146 T^{2} - 1074 T^{3} + 6914 T^{4} - 1074 p T^{5} + 146 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 42 T^{2} - 192 T^{3} + 151 T^{4} - 192 p T^{5} + 42 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 22 T + 242 T^{2} + 1662 T^{3} + 10346 T^{4} + 1662 p T^{5} + 242 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 75 T^{2} + 462 T^{3} + 2558 T^{4} + 462 p T^{5} + 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 75 T^{2} - 66 T^{3} + 1834 T^{4} - 66 p T^{5} + 75 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 115 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 92 T^{2} + 228 T^{3} + 2630 T^{4} + 228 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 272 T^{2} + 1560 T^{3} + 8526 T^{4} + 1560 p T^{5} + 272 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 28 T + 405 T^{2} - 3744 T^{3} + 31000 T^{4} - 3744 p T^{5} + 405 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 206 T^{2} + 18 T^{3} + 18866 T^{4} + 18 p T^{5} + 206 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 254 T^{2} - 990 T^{3} + 28314 T^{4} - 990 p T^{5} + 254 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + p T^{2} - 1472 T^{3} + 11344 T^{4} - 1472 p T^{5} + p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 274 T^{2} + 2782 T^{3} + 34306 T^{4} + 2782 p T^{5} + 274 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.55917499103403814837117706331, −5.22811335731216469681248623343, −5.16408713449294106429679578114, −5.14117198837266932656179427163, −5.02676542878284103842518043036, −4.55073464665465719991411251167, −4.47701478353569070213202650668, −4.28929082997473489712410051355, −4.05518727827947949853236632678, −3.83214022093776075494272712468, −3.68652490452323096534963332410, −3.63682327774320897440643295655, −3.26072178120698813808410193441, −3.05674857242172317035079996890, −2.91640515334168227146497924438, −2.84510866519133220429922463080, −2.77076629632220862226554664254, −1.97389227441624471752744031581, −1.81212268695987907451131134791, −1.69225860155923813715176337943, −1.65504576928254545675711145381, −0.887501263415251068237805751718, −0.815774673367089226528461573546, −0.62488902432075305707320730903, −0.28019659636436873962703866560,
0.28019659636436873962703866560, 0.62488902432075305707320730903, 0.815774673367089226528461573546, 0.887501263415251068237805751718, 1.65504576928254545675711145381, 1.69225860155923813715176337943, 1.81212268695987907451131134791, 1.97389227441624471752744031581, 2.77076629632220862226554664254, 2.84510866519133220429922463080, 2.91640515334168227146497924438, 3.05674857242172317035079996890, 3.26072178120698813808410193441, 3.63682327774320897440643295655, 3.68652490452323096534963332410, 3.83214022093776075494272712468, 4.05518727827947949853236632678, 4.28929082997473489712410051355, 4.47701478353569070213202650668, 4.55073464665465719991411251167, 5.02676542878284103842518043036, 5.14117198837266932656179427163, 5.16408713449294106429679578114, 5.22811335731216469681248623343, 5.55917499103403814837117706331
