| L(s) = 1 | + 256·2-s − 1.84e3·3-s + 4.91e4·4-s + 1.56e5·5-s − 4.72e5·6-s − 9.84e5·7-s + 8.38e6·8-s − 2.19e6·9-s + 4.00e7·10-s + 1.11e8·11-s − 9.06e7·12-s + 2.89e8·13-s − 2.52e8·14-s − 2.88e8·15-s + 1.34e9·16-s + 1.42e9·17-s − 5.60e8·18-s + 6.15e9·19-s + 7.68e9·20-s + 1.81e9·21-s + 2.85e10·22-s − 4.33e9·23-s − 1.54e10·24-s + 1.83e10·25-s + 7.40e10·26-s − 1.21e10·27-s − 4.84e10·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.486·3-s + 3/2·4-s + 0.894·5-s − 0.688·6-s − 0.452·7-s + 1.41·8-s − 0.152·9-s + 1.26·10-s + 1.72·11-s − 0.730·12-s + 1.27·13-s − 0.639·14-s − 0.435·15-s + 5/4·16-s + 0.840·17-s − 0.215·18-s + 1.58·19-s + 1.34·20-s + 0.220·21-s + 2.44·22-s − 0.265·23-s − 0.688·24-s + 3/5·25-s + 1.80·26-s − 0.222·27-s − 0.678·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(7.802149427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.802149427\) |
| \(L(\frac{17}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 1844 T + 207074 p^{3} T^{2} + 1844 p^{15} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 984932 T + 778723088706 p T^{2} + 984932 p^{15} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 10141104 p T + 728383844593826 p T^{2} - 10141104 p^{16} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22254892 p T + 614040129797022 p^{2} T^{2} - 22254892 p^{16} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1421739348 T + 6225862099428528262 T^{2} - 1421739348 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6159406120 T + 39410443325042817798 T^{2} - 6159406120 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4330165884 T - \)\(11\!\cdots\!22\)\( T^{2} + 4330165884 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 164295941940 T + \)\(14\!\cdots\!98\)\( T^{2} + 164295941940 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 282710965016 T + \)\(46\!\cdots\!66\)\( T^{2} + 282710965016 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 790105159228 T + \)\(73\!\cdots\!82\)\( T^{2} - 790105159228 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 374717265276 T + \)\(19\!\cdots\!46\)\( T^{2} + 374717265276 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 923824433204 T + \)\(15\!\cdots\!26\)\( p T^{2} + 923824433204 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4796717212428 T + \)\(21\!\cdots\!82\)\( T^{2} - 4796717212428 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2768921292084 T - \)\(84\!\cdots\!22\)\( T^{2} + 2768921292084 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20737233989880 T + \)\(77\!\cdots\!22\)\( p T^{2} + 20737233989880 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 577887725164 T + \)\(97\!\cdots\!26\)\( T^{2} - 577887725164 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 86553258077668 T + \)\(63\!\cdots\!42\)\( T^{2} - 86553258077668 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 73838906689464 T + \)\(28\!\cdots\!26\)\( T^{2} - 73838906689464 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 39973727021476 T + \)\(12\!\cdots\!58\)\( T^{2} - 39973727021476 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 421665304874800 T + \)\(10\!\cdots\!98\)\( T^{2} - 421665304874800 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 721146660038964 T + \)\(24\!\cdots\!38\)\( T^{2} + 721146660038964 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 363712836623220 T + \)\(24\!\cdots\!98\)\( T^{2} - 363712836623220 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 289030099396948 T + \)\(11\!\cdots\!62\)\( T^{2} - 289030099396948 p^{15} T^{3} + p^{30} T^{4} \) |
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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
−16.92643205475262515583626570643, −16.77683200436030348164112878375, −16.01719778423616035572737560116, −15.06216042430586010407602211658, −14.23529056334680800000275933123, −13.92042686494785765624038293957, −13.04406072003805975213607098353, −12.43223860759449504643554059912, −11.40073356157614210382640980135, −11.17589325978277518626815501111, −9.777170891710757936159813134059, −9.174544246965007758659248960166, −7.53656591567008018063511862389, −6.51792103893196097256117846195, −5.87007184809743479809345747410, −5.34866802695151667810779327679, −3.84406424235310651912738150051, −3.37231254108050711152341656900, −1.82934783898564328314299555812, −1.05652376614415983558783477407,
1.05652376614415983558783477407, 1.82934783898564328314299555812, 3.37231254108050711152341656900, 3.84406424235310651912738150051, 5.34866802695151667810779327679, 5.87007184809743479809345747410, 6.51792103893196097256117846195, 7.53656591567008018063511862389, 9.174544246965007758659248960166, 9.777170891710757936159813134059, 11.17589325978277518626815501111, 11.40073356157614210382640980135, 12.43223860759449504643554059912, 13.04406072003805975213607098353, 13.92042686494785765624038293957, 14.23529056334680800000275933123, 15.06216042430586010407602211658, 16.01719778423616035572737560116, 16.77683200436030348164112878375, 16.92643205475262515583626570643
