| L(s) = 1 | + 12·7-s + 16·11-s − 48·13-s − 24·17-s − 56·23-s − 80·31-s − 112·37-s + 56·41-s + 8·43-s − 16·47-s + 72·49-s − 120·53-s − 24·61-s + 8·67-s − 272·71-s − 108·73-s + 192·77-s + 272·83-s − 576·91-s + 148·97-s − 152·101-s − 124·103-s − 160·107-s − 144·113-s − 288·119-s − 312·121-s + 127-s + ⋯ |
| L(s) = 1 | + 12/7·7-s + 1.45·11-s − 3.69·13-s − 1.41·17-s − 2.43·23-s − 2.58·31-s − 3.02·37-s + 1.36·41-s + 8/43·43-s − 0.340·47-s + 1.46·49-s − 2.26·53-s − 0.393·61-s + 8/67·67-s − 3.83·71-s − 1.47·73-s + 2.49·77-s + 3.27·83-s − 6.32·91-s + 1.52·97-s − 1.50·101-s − 1.20·103-s − 1.49·107-s − 1.27·113-s − 2.42·119-s − 2.57·121-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0001095756455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0001095756455\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6014 T^{4} - 660 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 252 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 21360 T^{3} + 319874 T^{4} + 21360 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} - 5448 T^{3} - 163198 T^{4} - 5448 p^{2} T^{5} + 288 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1004 T^{2} + 509190 T^{4} - 1004 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 45528 T^{3} + 1241282 T^{4} + 45528 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1424 T^{2} + 980610 T^{4} - 1424 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 40 T + 1722 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 112 T + 6272 T^{2} + 316848 T^{3} + 13874882 T^{4} + 316848 p^{2} T^{5} + 6272 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 28 T + 2382 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 3960 T^{3} - 5004286 T^{4} + 3960 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} - 33456 T^{3} - 9745438 T^{4} - 33456 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 120 T + 7200 T^{2} + 409080 T^{3} + 22882562 T^{4} + 409080 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8600 T^{2} + 42555378 T^{4} - 8600 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 4574 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} + 75000 T^{3} - 16429246 T^{4} + 75000 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 136 T + 14322 T^{2} + 136 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 108 T + 5832 T^{2} + 214596 T^{3} - 3272626 T^{4} + 214596 p^{2} T^{5} + 5832 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 556 T^{2} + 67139430 T^{4} + 556 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T + 36992 T^{2} - 3994320 T^{3} + 370520834 T^{4} - 3994320 p^{2} T^{5} + 36992 p^{4} T^{6} - 272 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23804 T^{2} + 265665990 T^{4} - 23804 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 74 T + 2738 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.46511046855768946764695522765, −6.18048795735791672903430111650, −6.17214805102423615065004134341, −5.62297563679052352177567679685, −5.42454499159831172182739244489, −5.24701295621894428009259561947, −5.21794485582512507503253009708, −4.77165234115165595819121431820, −4.58930637426654127748575375620, −4.57631691218532036584591948955, −4.48000389470065501016256752998, −3.79087105233315119668194643659, −3.74537719624819617674944962489, −3.71159077621917873505750563959, −3.53095043793525641951449354192, −2.62344616557468518279611848141, −2.54456828132656011166500781762, −2.48431590773195886236978996586, −2.37805854306248123029111828891, −1.75463237958958950307720057065, −1.58278448050718673906226123206, −1.39276645989792265415420596736, −1.37093775312197199164667815682, −0.04976566118830241009351476438, −0.01308342387715490128779479009,
0.01308342387715490128779479009, 0.04976566118830241009351476438, 1.37093775312197199164667815682, 1.39276645989792265415420596736, 1.58278448050718673906226123206, 1.75463237958958950307720057065, 2.37805854306248123029111828891, 2.48431590773195886236978996586, 2.54456828132656011166500781762, 2.62344616557468518279611848141, 3.53095043793525641951449354192, 3.71159077621917873505750563959, 3.74537719624819617674944962489, 3.79087105233315119668194643659, 4.48000389470065501016256752998, 4.57631691218532036584591948955, 4.58930637426654127748575375620, 4.77165234115165595819121431820, 5.21794485582512507503253009708, 5.24701295621894428009259561947, 5.42454499159831172182739244489, 5.62297563679052352177567679685, 6.17214805102423615065004134341, 6.18048795735791672903430111650, 6.46511046855768946764695522765
