| L(s) = 1 | + 9.60e3·7-s − 2.48e4·13-s − 6.11e5·19-s − 4.93e6·25-s − 4.76e6·31-s − 4.65e6·37-s + 7.16e4·43-s + 5.76e7·49-s + 2.36e7·61-s − 3.02e7·67-s − 3.59e8·73-s − 2.11e8·79-s − 2.38e8·91-s − 1.39e9·97-s − 8.03e8·103-s − 2.57e9·109-s − 7.12e9·121-s + 127-s + 131-s − 5.87e9·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.240·13-s − 1.07·19-s − 2.52·25-s − 0.926·31-s − 0.407·37-s + 0.00319·43-s + 10/7·49-s + 0.219·61-s − 0.183·67-s − 1.48·73-s − 0.610·79-s − 0.364·91-s − 1.59·97-s − 0.703·103-s − 1.74·109-s − 3.02·121-s − 1.62·133-s − 3.42·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 4937684 T^{2} + 498503834358 p^{2} T^{4} + 4937684 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 7126924172 T^{2} + 23799823062061527942 T^{4} + 7126924172 p^{18} T^{6} + p^{36} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 12404 T + 18385832046 T^{2} + 12404 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 222697786244 T^{2} + \)\(37\!\cdots\!26\)\( T^{4} + 222697786244 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 305648 T + 185114846358 T^{2} + 305648 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 5088616816028 T^{2} + \)\(12\!\cdots\!78\)\( T^{4} + 5088616816028 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 9668196527692 T^{2} + \)\(10\!\cdots\!82\)\( T^{4} - 9668196527692 p^{18} T^{6} + p^{36} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2382128 T + 52211748407022 T^{2} + 2382128 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 62852 p T + 5693407983366 p T^{2} + 62852 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 1101660868624676 T^{2} + \)\(50\!\cdots\!62\)\( T^{4} + 1101660868624676 p^{18} T^{6} + p^{36} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 35800 T + 15793621564710 T^{2} - 35800 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 501361251370564 T^{2} + \)\(19\!\cdots\!46\)\( T^{4} - 501361251370564 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 3096207687875116 T^{2} + \)\(77\!\cdots\!78\)\( T^{4} - 3096207687875116 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 457588613861380 p T^{2} + \)\(31\!\cdots\!86\)\( T^{4} + 457588613861380 p^{19} T^{6} + p^{36} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 11842852 T + 11712045554108142 T^{2} - 11842852 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 15135416 T + 33578128266033942 T^{2} + 15135416 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 145680435773216348 T^{2} + \)\(93\!\cdots\!02\)\( T^{4} + 145680435773216348 p^{18} T^{6} + p^{36} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 179910164 T + 95534659227954486 T^{2} + 179910164 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 105729248 T + 96040372764525150 T^{2} + 105729248 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 586701377890551500 T^{2} + \)\(15\!\cdots\!54\)\( T^{4} + 586701377890551500 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 673820235643606628 T^{2} + \)\(25\!\cdots\!34\)\( T^{4} + 673820235643606628 p^{18} T^{6} + p^{36} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 7175252 p T + 1192287202224854118 T^{2} + 7175252 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
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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
−7.84325205930893630979574996291, −7.21188387321442611401531609666, −7.20483776147752814518579797765, −7.08957260250366716700854055271, −6.68679474906360266129142233023, −6.11824272759831586376077894312, −5.97758217096317328632915079287, −5.92601747288007003907385101983, −5.68580141873681230815787385611, −5.11751850231862559693427468963, −4.90075188794546422627929611371, −4.73864944530868316881834853886, −4.71478191077425634097165749943, −3.84345235300561395214118493610, −3.80444872580990799293474670440, −3.80199776178950117113745991936, −3.66241565838291852937799501491, −2.64930590323901791696543482516, −2.50679989188284260110175031457, −2.47416047688211414760179224595, −2.20594064020517275728257795978, −1.58845429098088371097031215141, −1.36118544905254006457235137927, −1.26887277495036958972612226857, −1.13417535806653471803286523279, 0, 0, 0, 0,
1.13417535806653471803286523279, 1.26887277495036958972612226857, 1.36118544905254006457235137927, 1.58845429098088371097031215141, 2.20594064020517275728257795978, 2.47416047688211414760179224595, 2.50679989188284260110175031457, 2.64930590323901791696543482516, 3.66241565838291852937799501491, 3.80199776178950117113745991936, 3.80444872580990799293474670440, 3.84345235300561395214118493610, 4.71478191077425634097165749943, 4.73864944530868316881834853886, 4.90075188794546422627929611371, 5.11751850231862559693427468963, 5.68580141873681230815787385611, 5.92601747288007003907385101983, 5.97758217096317328632915079287, 6.11824272759831586376077894312, 6.68679474906360266129142233023, 7.08957260250366716700854055271, 7.20483776147752814518579797765, 7.21188387321442611401531609666, 7.84325205930893630979574996291
