| L(s) = 1 | + 4.07e3·3-s − 1.40e5·5-s − 1.26e5·7-s + 7.49e6·9-s − 2.06e7·11-s + 4.99e8·13-s − 5.71e8·15-s + 3.13e9·17-s + 4.74e8·19-s − 5.13e8·21-s + 4.07e10·23-s − 1.54e10·25-s + 5.19e10·27-s + 3.52e10·29-s + 3.43e10·31-s − 8.42e10·33-s + 1.76e10·35-s + 8.70e11·37-s + 2.03e12·39-s + 9.00e11·41-s − 5.00e11·43-s − 1.05e12·45-s − 1.20e12·47-s − 9.29e12·49-s + 1.27e13·51-s + 1.23e12·53-s + 2.90e12·55-s + ⋯ |
| L(s) = 1 | + 1.07·3-s − 0.802·5-s − 0.0579·7-s + 0.522·9-s − 0.320·11-s + 2.20·13-s − 0.863·15-s + 1.85·17-s + 0.121·19-s − 0.0622·21-s + 2.49·23-s − 0.507·25-s + 0.955·27-s + 0.379·29-s + 0.224·31-s − 0.344·33-s + 0.0465·35-s + 1.50·37-s + 2.37·39-s + 0.721·41-s − 0.280·43-s − 0.419·45-s − 0.347·47-s − 1.95·49-s + 1.99·51-s + 0.144·53-s + 0.256·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(4.845405293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.845405293\) |
| \(L(\frac{17}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 4072 T + 336530 p^{3} T^{2} - 4072 p^{15} T^{3} + p^{30} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 28052 p T + 1406582414 p^{2} T^{2} + 28052 p^{16} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 126192 T + 1329524863586 p T^{2} + 126192 p^{15} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 20682632 T + 122319220218178 p T^{2} + 20682632 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 38446652 p T + 974516755366782 p^{2} T^{2} - 38446652 p^{16} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3139516900 T + 7453966489546885286 T^{2} - 3139516900 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 474668552 T + 28477910697117701574 T^{2} - 474668552 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 40776002608 T + \)\(94\!\cdots\!30\)\( T^{2} - 40776002608 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 35253157356 T + \)\(85\!\cdots\!82\)\( T^{2} - 35253157356 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 34389193280 T + \)\(24\!\cdots\!02\)\( T^{2} - 34389193280 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 870228564444 T + \)\(75\!\cdots\!70\)\( T^{2} - 870228564444 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 900085452084 T - \)\(19\!\cdots\!34\)\( T^{2} - 900085452084 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 500707998536 T + \)\(30\!\cdots\!38\)\( T^{2} + 500707998536 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1208059119264 T + \)\(21\!\cdots\!10\)\( T^{2} + 1208059119264 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1236734202044 T + \)\(11\!\cdots\!98\)\( T^{2} - 1236734202044 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14441975905064 T + \)\(76\!\cdots\!22\)\( T^{2} + 14441975905064 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18336303417260 T + \)\(12\!\cdots\!02\)\( T^{2} - 18336303417260 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 76601421514856 T + \)\(47\!\cdots\!70\)\( T^{2} - 76601421514856 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 145877173886864 T + \)\(16\!\cdots\!26\)\( T^{2} - 145877173886864 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 26417269924108 T + \)\(10\!\cdots\!30\)\( T^{2} + 26417269924108 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 257907833388128 T + \)\(71\!\cdots\!94\)\( T^{2} + 257907833388128 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 255512806582648 T + \)\(10\!\cdots\!90\)\( T^{2} + 255512806582648 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 719794611712812 T + \)\(38\!\cdots\!34\)\( T^{2} + 719794611712812 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 407635590418756 T - \)\(29\!\cdots\!30\)\( T^{2} - 407635590418756 p^{15} T^{3} + p^{30} T^{4} \) |
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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
−15.64436018072153669198806343263, −15.14341375616834627869739487466, −14.35341159387234652448179250857, −13.91783694385469392075864340310, −12.94203236127974397383029872440, −12.71327040351165631588581608449, −11.34332697711734912293348437115, −11.17297734284009146202490202446, −10.04083493988207157462660641592, −9.282129562600171720234225893476, −8.282082556495118819925545708619, −8.210374870027580357505834850846, −7.21089659078651608761238311671, −6.23950975694699971103197654089, −5.19723294813748792731029957924, −4.06837200278077667317663998383, −3.27336660584394385911731943158, −2.90079594633906577563969189815, −1.31527183865820497371551946069, −0.846061202646158350681722241755,
0.846061202646158350681722241755, 1.31527183865820497371551946069, 2.90079594633906577563969189815, 3.27336660584394385911731943158, 4.06837200278077667317663998383, 5.19723294813748792731029957924, 6.23950975694699971103197654089, 7.21089659078651608761238311671, 8.210374870027580357505834850846, 8.282082556495118819925545708619, 9.282129562600171720234225893476, 10.04083493988207157462660641592, 11.17297734284009146202490202446, 11.34332697711734912293348437115, 12.71327040351165631588581608449, 12.94203236127974397383029872440, 13.91783694385469392075864340310, 14.35341159387234652448179250857, 15.14341375616834627869739487466, 15.64436018072153669198806343263
