| L(s) = 1 | − 4·2-s + 4·4-s − 4·5-s + 16·8-s + 25·9-s + 16·10-s − 42·13-s − 64·16-s − 14·17-s − 100·18-s − 16·20-s + 24·25-s + 168·26-s − 58·29-s + 64·32-s + 56·34-s + 100·36-s − 24·37-s − 64·40-s − 124·41-s − 100·45-s + 185·49-s − 96·50-s − 168·52-s − 2·53-s + 232·58-s − 128·61-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s − 4/5·5-s + 2·8-s + 25/9·9-s + 8/5·10-s − 3.23·13-s − 4·16-s − 0.823·17-s − 5.55·18-s − 4/5·20-s + 0.959·25-s + 6.46·26-s − 2·29-s + 2·32-s + 1.64·34-s + 25/9·36-s − 0.648·37-s − 8/5·40-s − 3.02·41-s − 2.22·45-s + 3.77·49-s − 1.91·50-s − 3.23·52-s − 0.0377·53-s + 4·58-s − 2.09·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33362176 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1989563350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1989563350\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 304 T^{4} - 25 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 2 T - 6 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 13344 T^{4} - 185 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 296 T^{2} + 45486 T^{4} - 296 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 21 T + 320 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 7 T + 576 T^{2} + 7 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + p T^{2} + 556608 T^{4} + p^{5} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + p T + 1536 T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1028 T^{2} + 1177350 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 1862 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 62 T + 2898 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4100 T^{2} + 8938854 T^{4} - 4100 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4772 T^{2} + 13351110 T^{4} - 4772 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + T + 4920 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5897 T^{2} + 29408496 T^{4} - 5897 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 64 T + 8238 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4297 T^{2} + 10638480 T^{4} - 4297 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 14552 T^{2} + 99101166 T^{4} - 14552 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 169 T + 17100 T^{2} + 169 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1112 T^{2} + 77304366 T^{4} - 1112 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5480 T^{2} - 19410066 T^{4} - 5480 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 13074 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 24 T + 7790 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−10.53204233362395266473197041057, −10.15156645051961577363883938392, −9.842156567191132062974688842440, −9.738576406944993851754355580447, −9.345268806069075198547626287937, −8.975746652778786052636048396828, −8.807327855303875482613790456286, −8.647610547000475939808125906103, −7.923104400566152651706234610668, −7.71010912006590510594584597752, −7.33163752798424649704836560897, −7.21418394368749186378275950445, −7.15091971872873401847872504131, −7.01629702535989184296324053586, −6.25004596022357815922488691365, −5.48063168836927258090270426209, −4.91431741519517579633433478371, −4.85240841450964680616579909879, −4.46101652016326801070332098062, −4.12567214299558682290006844227, −3.75955991938948054450110569198, −2.76573626904382258291628180877, −1.79739967030431962405015033792, −1.73985840038128703296227782032, −0.38075786261024034835814460205,
0.38075786261024034835814460205, 1.73985840038128703296227782032, 1.79739967030431962405015033792, 2.76573626904382258291628180877, 3.75955991938948054450110569198, 4.12567214299558682290006844227, 4.46101652016326801070332098062, 4.85240841450964680616579909879, 4.91431741519517579633433478371, 5.48063168836927258090270426209, 6.25004596022357815922488691365, 7.01629702535989184296324053586, 7.15091971872873401847872504131, 7.21418394368749186378275950445, 7.33163752798424649704836560897, 7.71010912006590510594584597752, 7.923104400566152651706234610668, 8.647610547000475939808125906103, 8.807327855303875482613790456286, 8.975746652778786052636048396828, 9.345268806069075198547626287937, 9.738576406944993851754355580447, 9.842156567191132062974688842440, 10.15156645051961577363883938392, 10.53204233362395266473197041057
