| L(s) = 1 | − 3.93e4·3-s − 2.32e6·5-s + 2.79e7·7-s + 1.16e9·9-s + 4.90e9·11-s + 5.28e10·13-s + 9.16e10·15-s − 5.30e11·17-s + 4.01e11·19-s − 1.09e12·21-s + 9.62e12·23-s − 3.35e13·25-s − 3.05e13·27-s + 9.81e12·29-s − 9.55e13·31-s − 1.92e14·33-s − 6.49e13·35-s + 2.86e14·37-s − 2.08e15·39-s − 2.41e15·41-s + 1.71e14·43-s − 2.70e15·45-s + 4.57e15·47-s − 2.01e16·49-s + 2.08e16·51-s − 4.80e16·53-s − 1.14e16·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.532·5-s + 0.261·7-s + 9-s + 0.626·11-s + 1.38·13-s + 0.615·15-s − 1.08·17-s + 0.285·19-s − 0.301·21-s + 1.11·23-s − 1.75·25-s − 0.769·27-s + 0.125·29-s − 0.648·31-s − 0.723·33-s − 0.139·35-s + 0.361·37-s − 1.59·39-s − 1.15·41-s + 0.0520·43-s − 0.532·45-s + 0.596·47-s − 1.76·49-s + 1.25·51-s − 1.99·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(2.119330535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119330535\) |
| \(L(\frac{21}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{9} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 465428 p T + 62287472414 p^{4} T^{2} + 465428 p^{20} T^{3} + p^{38} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 27922992 T + 2992869783111458 p T^{2} - 27922992 p^{19} T^{3} + p^{38} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4900532536 T - 3341988404603682398 p T^{2} - 4900532536 p^{19} T^{3} + p^{38} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 52865932268 T + \)\(10\!\cdots\!02\)\( p T^{2} - 52865932268 p^{19} T^{3} + p^{38} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 31190518684 p T + \)\(78\!\cdots\!18\)\( p^{2} T^{2} + 31190518684 p^{20} T^{3} + p^{38} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 401191851080 T + \)\(26\!\cdots\!22\)\( T^{2} - 401191851080 p^{19} T^{3} + p^{38} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9624459262960 T + \)\(15\!\cdots\!58\)\( T^{2} - 9624459262960 p^{19} T^{3} + p^{38} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9815300727660 T + \)\(87\!\cdots\!82\)\( T^{2} - 9815300727660 p^{19} T^{3} + p^{38} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 95520008990368 T + \)\(44\!\cdots\!82\)\( T^{2} + 95520008990368 p^{19} T^{3} + p^{38} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 286149355418556 T + \)\(68\!\cdots\!30\)\( T^{2} - 286149355418556 p^{19} T^{3} + p^{38} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2413041231470796 T + \)\(84\!\cdots\!62\)\( T^{2} + 2413041231470796 p^{19} T^{3} + p^{38} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 171463305783544 T + \)\(11\!\cdots\!98\)\( T^{2} - 171463305783544 p^{19} T^{3} + p^{38} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4576185209446944 T + \)\(56\!\cdots\!46\)\( T^{2} - 4576185209446944 p^{19} T^{3} + p^{38} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 48034770833136004 T + \)\(17\!\cdots\!94\)\( T^{2} + 48034770833136004 p^{19} T^{3} + p^{38} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 55665234820264024 T + \)\(82\!\cdots\!38\)\( T^{2} - 55665234820264024 p^{19} T^{3} + p^{38} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 159404241162019636 T + \)\(22\!\cdots\!50\)\( T^{2} + 159404241162019636 p^{19} T^{3} + p^{38} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 416133498234756200 T + \)\(13\!\cdots\!22\)\( T^{2} - 416133498234756200 p^{19} T^{3} + p^{38} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1198585214434221392 T + \)\(64\!\cdots\!02\)\( T^{2} - 1198585214434221392 p^{19} T^{3} + p^{38} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 721645320236267212 T + \)\(61\!\cdots\!14\)\( T^{2} + 721645320236267212 p^{19} T^{3} + p^{38} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1980034735036691008 T + \)\(25\!\cdots\!50\)\( T^{2} - 1980034735036691008 p^{19} T^{3} + p^{38} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1287426982220624072 T + \)\(59\!\cdots\!66\)\( T^{2} - 1287426982220624072 p^{19} T^{3} + p^{38} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 635691821543995116 T + \)\(19\!\cdots\!86\)\( T^{2} + 635691821543995116 p^{19} T^{3} + p^{38} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5880822596854316740 T + \)\(12\!\cdots\!70\)\( T^{2} - 5880822596854316740 p^{19} T^{3} + p^{38} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.73210642202552607997695874621, −11.52127384963938859357509161584, −10.98375926180862205218050313426, −10.76619979563221245367481303833, −9.671830586274068961371351605242, −9.381955795014691417460229632316, −8.507817699532738437261174686345, −8.026402365151211121871400087063, −7.29839610421003992816125190128, −6.58803258649418832511621950532, −6.28020484244533726207863524844, −5.61802506105730960303797765232, −4.84626378675259786094492902647, −4.45521236970188104494381735097, −3.59231961671828742528656662177, −3.37045881279640315036724866728, −1.91262919461301695658677376410, −1.73391080765629999868629411490, −0.72557930526593880852018609870, −0.49492852262120571557703507951,
0.49492852262120571557703507951, 0.72557930526593880852018609870, 1.73391080765629999868629411490, 1.91262919461301695658677376410, 3.37045881279640315036724866728, 3.59231961671828742528656662177, 4.45521236970188104494381735097, 4.84626378675259786094492902647, 5.61802506105730960303797765232, 6.28020484244533726207863524844, 6.58803258649418832511621950532, 7.29839610421003992816125190128, 8.026402365151211121871400087063, 8.507817699532738437261174686345, 9.381955795014691417460229632316, 9.671830586274068961371351605242, 10.76619979563221245367481303833, 10.98375926180862205218050313426, 11.52127384963938859357509161584, 11.73210642202552607997695874621
