| L(s) = 1 | − 6·3-s − 4·4-s − 2·5-s − 10·7-s + 17·9-s + 24·12-s − 8·13-s + 12·15-s + 12·16-s − 12·17-s + 8·20-s + 60·21-s + 6·23-s + 25-s − 30·27-s + 40·28-s − 8·31-s + 20·35-s − 68·36-s − 12·37-s + 48·39-s − 28·43-s − 34·45-s + 12·47-s − 72·48-s + 61·49-s + 72·51-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 2·4-s − 0.894·5-s − 3.77·7-s + 17/3·9-s + 6.92·12-s − 2.21·13-s + 3.09·15-s + 3·16-s − 2.91·17-s + 1.78·20-s + 13.0·21-s + 1.25·23-s + 1/5·25-s − 5.77·27-s + 7.55·28-s − 1.43·31-s + 3.38·35-s − 11.3·36-s − 1.97·37-s + 7.68·39-s − 4.26·43-s − 5.06·45-s + 1.75·47-s − 10.3·48-s + 61/7·49-s + 10.0·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2463 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 20 T^{2} + 39 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 246 T^{3} + 1428 T^{4} - 246 p T^{5} + 53 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 38 T^{2} + 1443 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T - 8 T^{2} + 80 T^{3} + 2239 T^{4} + 80 p T^{5} - 8 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 49 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 4707 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 3 p T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 164 T^{2} + 1392 T^{3} + 13191 T^{4} + 1392 p T^{5} + 164 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 128 T^{2} + 960 T^{3} + 5751 T^{4} + 960 p T^{5} + 128 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 14 T + 79 T^{2} - 70 T^{3} - 1988 T^{4} - 70 p T^{5} + 79 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 5 T^{2} - 290 T^{3} - 164 T^{4} - 290 p T^{5} - 5 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 24 T + 368 T^{2} + 4224 T^{3} + 39663 T^{4} + 4224 p T^{5} + 368 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T + 200 T^{2} + 1824 T^{3} + 20655 T^{4} + 1824 p T^{5} + 200 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 178 T^{2} + 16299 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 281 T^{2} - 3114 T^{3} + 31620 T^{4} - 3114 p T^{5} + 281 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 27462 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−9.281772559600105749965630978990, −9.125906448384533328275340505616, −8.791461889856836695613114174551, −8.482326698885140156043849878094, −8.340101778864542092392593481380, −7.53293967222389299818625715395, −7.34801701601554728053257820494, −7.19732660168812778337160342315, −6.90435500159865840459380640733, −6.79105016565175011509637243879, −6.50948641537538032316247065784, −5.98519094135951549798139863897, −5.94785832514809899906644857258, −5.94408069937196695075850119713, −5.42183200585615445833258153681, −4.97530387522793433700803239173, −4.90601835855517239878397692751, −4.66926394236766159527419242616, −4.48135591222057677570545678753, −4.00471873064514068208803733396, −3.46970536384755394067781227721, −3.45844640485722810805123464210, −3.00268188933097995799999720595, −2.53422944431611399476909669148, −1.48247515813452362094316225641, 0, 0, 0, 0,
1.48247515813452362094316225641, 2.53422944431611399476909669148, 3.00268188933097995799999720595, 3.45844640485722810805123464210, 3.46970536384755394067781227721, 4.00471873064514068208803733396, 4.48135591222057677570545678753, 4.66926394236766159527419242616, 4.90601835855517239878397692751, 4.97530387522793433700803239173, 5.42183200585615445833258153681, 5.94408069937196695075850119713, 5.94785832514809899906644857258, 5.98519094135951549798139863897, 6.50948641537538032316247065784, 6.79105016565175011509637243879, 6.90435500159865840459380640733, 7.19732660168812778337160342315, 7.34801701601554728053257820494, 7.53293967222389299818625715395, 8.340101778864542092392593481380, 8.482326698885140156043849878094, 8.791461889856836695613114174551, 9.125906448384533328275340505616, 9.281772559600105749965630978990
