| L(s) = 1 | + 4.37e3·3-s + 2.91e5·5-s − 1.16e6·7-s + 1.43e7·9-s − 2.98e7·11-s − 3.86e8·13-s + 1.27e9·15-s − 2.20e9·17-s − 2.78e9·19-s − 5.07e9·21-s − 4.41e10·23-s + 8.50e9·25-s + 4.18e10·27-s + 1.32e11·29-s − 5.93e10·31-s − 1.30e11·33-s − 3.38e11·35-s + 6.64e11·37-s − 1.68e12·39-s + 2.14e11·41-s − 3.40e12·43-s + 4.18e12·45-s − 6.89e12·47-s + 5.56e12·49-s − 9.63e12·51-s + 4.38e12·53-s − 8.68e12·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.66·5-s − 0.532·7-s + 9-s − 0.461·11-s − 1.70·13-s + 1.92·15-s − 1.30·17-s − 0.715·19-s − 0.615·21-s − 2.70·23-s + 0.278·25-s + 0.769·27-s + 1.42·29-s − 0.387·31-s − 0.532·33-s − 0.888·35-s + 1.15·37-s − 1.97·39-s + 0.171·41-s − 1.90·43-s + 1.66·45-s − 1.98·47-s + 1.17·49-s − 1.50·51-s + 0.512·53-s − 0.769·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 291404 T + 15282775078 p T^{2} - 291404 p^{15} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 165888 p T - 86009601586 p^{2} T^{2} + 165888 p^{16} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 29802008 T + 152422959012034 p T^{2} + 29802008 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 29695972 p T + 448771484454078 p^{2} T^{2} + 29695972 p^{16} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2203297340 T + 6584909326821682022 T^{2} + 2203297340 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2786896936 T + 26116147797810450246 T^{2} + 2786896936 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 44181695888 T + \)\(97\!\cdots\!14\)\( T^{2} + 44181695888 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 132679322364 T + \)\(20\!\cdots\!22\)\( T^{2} - 132679322364 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 59353215472 T - \)\(11\!\cdots\!78\)\( T^{2} + 59353215472 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 664408938396 T + \)\(77\!\cdots\!66\)\( T^{2} - 664408938396 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 214365009876 T + \)\(84\!\cdots\!70\)\( T^{2} - 214365009876 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3402660367640 T + \)\(80\!\cdots\!98\)\( T^{2} + 3402660367640 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6898255589376 T + \)\(34\!\cdots\!66\)\( T^{2} + 6898255589376 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4385018130188 T + \)\(15\!\cdots\!94\)\( T^{2} - 4385018130188 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14002837619864 T + \)\(65\!\cdots\!06\)\( T^{2} + 14002837619864 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 32462344447444 T + \)\(12\!\cdots\!30\)\( T^{2} + 32462344447444 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 56246702804392 T + \)\(55\!\cdots\!58\)\( T^{2} + 56246702804392 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 70013276735344 T + \)\(66\!\cdots\!50\)\( T^{2} + 70013276735344 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 162035381041036 T + \)\(23\!\cdots\!38\)\( T^{2} + 162035381041036 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14036765116144 T + \)\(38\!\cdots\!82\)\( T^{2} - 14036765116144 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 335133794517464 T + \)\(14\!\cdots\!02\)\( T^{2} - 335133794517464 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 213386440764396 T + \)\(32\!\cdots\!58\)\( T^{2} + 213386440764396 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 260434884895100 T + \)\(94\!\cdots\!10\)\( T^{2} + 260434884895100 p^{15} T^{3} + p^{30} T^{4} \) |
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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
−12.17451871996495937285455420044, −11.93206399357366146455585816388, −10.58806570420391508600707819966, −10.20671438347318920350201327237, −9.619568634274423868720070655079, −9.576692328197917291548554210349, −8.646344886691128314703819852448, −8.071023237415311415215785422428, −7.46585711873408505642060706661, −6.50808420054470951081214702446, −6.26400545698446715220249684843, −5.42697157539113225570862038138, −4.54074805913151669829380282689, −4.10710817491153812694886142018, −2.85799249807766849324776019629, −2.60479236154914226581716645621, −1.84473098431188223638036853705, −1.72909716236441631944643437423, 0, 0,
1.72909716236441631944643437423, 1.84473098431188223638036853705, 2.60479236154914226581716645621, 2.85799249807766849324776019629, 4.10710817491153812694886142018, 4.54074805913151669829380282689, 5.42697157539113225570862038138, 6.26400545698446715220249684843, 6.50808420054470951081214702446, 7.46585711873408505642060706661, 8.071023237415311415215785422428, 8.646344886691128314703819852448, 9.576692328197917291548554210349, 9.619568634274423868720070655079, 10.20671438347318920350201327237, 10.58806570420391508600707819966, 11.93206399357366146455585816388, 12.17451871996495937285455420044
