| L(s) = 1 | − 128·4-s + 60·5-s + 980·9-s + 1.78e4·11-s + 1.22e4·16-s + 6.36e4·19-s − 7.68e3·20-s + 4.48e4·25-s + 1.69e5·29-s − 3.94e5·31-s − 1.25e5·36-s + 2.32e5·41-s − 2.27e6·44-s + 5.88e4·45-s + 2.90e6·49-s + 1.06e6·55-s + 2.09e6·59-s − 5.25e6·61-s − 1.04e6·64-s − 7.83e6·71-s − 8.14e6·76-s + 7.72e6·79-s + 7.37e5·80-s + 8.75e5·81-s − 3.34e7·89-s + 3.81e6·95-s + 1.74e7·99-s + ⋯ |
| L(s) = 1 | − 4-s + 0.214·5-s + 0.448·9-s + 4.03·11-s + 3/4·16-s + 2.12·19-s − 0.214·20-s + 0.574·25-s + 1.29·29-s − 2.37·31-s − 0.448·36-s + 0.526·41-s − 4.03·44-s + 0.0961·45-s + 3.53·49-s + 0.865·55-s + 1.32·59-s − 2.96·61-s − 1/2·64-s − 2.59·71-s − 2.12·76-s + 1.76·79-s + 0.160·80-s + 0.183·81-s − 5.03·89-s + 0.456·95-s + 1.80·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.923520611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923520611\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 12 p T - 66 p^{4} T^{2} - 12 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 980 T^{2} + 9382 p^{2} T^{4} - 980 p^{14} T^{6} + p^{28} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2907140 T^{2} + 3444026008998 T^{4} - 2907140 p^{14} T^{6} + p^{28} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8904 T + 58001046 T^{2} - 8904 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 207134260 T^{2} + 18369334894869078 T^{4} - 207134260 p^{14} T^{6} + p^{28} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 513791940 T^{2} + 236061419749305158 T^{4} - 513791940 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 31800 T + 833487878 T^{2} - 31800 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6583044420 T^{2} + 22546653652650262118 T^{4} - 6583044420 p^{14} T^{6} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 84780 T + 15469426318 T^{2} - 84780 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 197056 T + 59904732606 T^{2} + 197056 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 238521132500 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - 238521132500 p^{14} T^{6} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 116244 T + 205827060246 T^{2} - 116244 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1046615537780 T^{2} + \)\(42\!\cdots\!98\)\( T^{4} - 1046615537780 p^{14} T^{6} + p^{28} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 115388894940 T^{2} - \)\(27\!\cdots\!62\)\( T^{4} + 115388894940 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1677817211540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - 1677817211540 p^{14} T^{6} + p^{28} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 1046760 T + 4817827968438 T^{2} - 1046760 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2625716 T + 8002437582606 T^{2} + 2625716 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16580251270100 T^{2} + \)\(13\!\cdots\!58\)\( T^{4} - 16580251270100 p^{14} T^{6} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3916176 T + 13255881578926 T^{2} + 3916176 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 37988250868580 T^{2} + \)\(59\!\cdots\!18\)\( T^{4} - 37988250868580 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 3863520 T + 34443978644318 T^{2} - 3863520 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 7805733448980 T^{2} + \)\(31\!\cdots\!58\)\( T^{4} - 7805733448980 p^{14} T^{6} + p^{28} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 16735020 T + 154740910351158 T^{2} + 16735020 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 161931097215620 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - 161931097215620 p^{14} T^{6} + p^{28} T^{8} \) |
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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
−14.21426954094864078302826728911, −13.99706425165724811076311947306, −13.92559472015997342315228999571, −13.40234642670442655734536262262, −12.60460992918555797720767998348, −12.39358215853472067479845814737, −12.04272751746887657229160282832, −11.47773087953961666413120642771, −11.43367107693412124962151191428, −10.45349184246193126009731803838, −10.14295639502063941940733858177, −9.317644992569716089361416705973, −9.249043236612006197813301497945, −9.039626562876661235970843121495, −8.557909576579256318904402123345, −7.33587238188566509601062922084, −7.30979794387154382333110495472, −6.56402003137759792453762556057, −5.93088416668753855450962986916, −5.32554859408978323330465615091, −4.20656131698610987747333475998, −4.09086863770160314906664794995, −3.25971690423980935490263412487, −1.37003475765328559695814707233, −1.05704594893896707814521876076,
1.05704594893896707814521876076, 1.37003475765328559695814707233, 3.25971690423980935490263412487, 4.09086863770160314906664794995, 4.20656131698610987747333475998, 5.32554859408978323330465615091, 5.93088416668753855450962986916, 6.56402003137759792453762556057, 7.30979794387154382333110495472, 7.33587238188566509601062922084, 8.557909576579256318904402123345, 9.039626562876661235970843121495, 9.249043236612006197813301497945, 9.317644992569716089361416705973, 10.14295639502063941940733858177, 10.45349184246193126009731803838, 11.43367107693412124962151191428, 11.47773087953961666413120642771, 12.04272751746887657229160282832, 12.39358215853472067479845814737, 12.60460992918555797720767998348, 13.40234642670442655734536262262, 13.92559472015997342315228999571, 13.99706425165724811076311947306, 14.21426954094864078302826728911
