| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.204574062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204574062\) |
| \(L(\frac{17}{2})\) |
|
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
−17.98743405112429646909491601924, −17.93141819570888901536804707999, −16.53119729755688797946361142611, −16.28672563384899712417247302567, −15.66908827177854488616155138602, −14.58320711481446394725029693025, −13.75529720571998686522206014545, −12.77475607150245383652151371922, −11.72765672763874089319535058503, −11.59420983772793419292327645457, −10.57462376958295016153351411067, −9.674250391342427018888484476418, −8.262384828822491896994491633370, −7.67793198394861137419437457875, −6.03372693114748030212806869321, −5.90039996848995902934037329616, −4.19861867695133090937414233451, −3.54756491121400270853524861383, −1.54117036825025129883850161570, −0.56824712043044151201863432086,
0.56824712043044151201863432086, 1.54117036825025129883850161570, 3.54756491121400270853524861383, 4.19861867695133090937414233451, 5.90039996848995902934037329616, 6.03372693114748030212806869321, 7.67793198394861137419437457875, 8.262384828822491896994491633370, 9.674250391342427018888484476418, 10.57462376958295016153351411067, 11.59420983772793419292327645457, 11.72765672763874089319535058503, 12.77475607150245383652151371922, 13.75529720571998686522206014545, 14.58320711481446394725029693025, 15.66908827177854488616155138602, 16.28672563384899712417247302567, 16.53119729755688797946361142611, 17.93141819570888901536804707999, 17.98743405112429646909491601924
