| L(s) = 1 | − 42·5-s − 136·7-s + 23·9-s − 222·11-s + 300·17-s − 114·19-s + 78·23-s − 1.36e3·25-s + 5.71e3·35-s − 2.98e3·43-s − 966·45-s + 7.57e3·47-s + 3.27e3·49-s + 9.32e3·55-s + 158·61-s − 3.12e3·63-s − 1.51e4·73-s + 3.01e4·77-s + 4.81e3·81-s − 3.32e4·83-s − 1.26e4·85-s + 4.78e3·95-s − 5.10e3·99-s − 1.31e4·101-s − 3.27e3·115-s − 4.08e4·119-s − 1.72e4·121-s + ⋯ |
| L(s) = 1 | − 1.67·5-s − 2.77·7-s + 0.283·9-s − 1.83·11-s + 1.03·17-s − 0.315·19-s + 0.147·23-s − 2.17·25-s + 4.66·35-s − 1.61·43-s − 0.477·45-s + 3.43·47-s + 1.36·49-s + 3.08·55-s + 0.0424·61-s − 0.788·63-s − 2.84·73-s + 5.09·77-s + 0.733·81-s − 4.83·83-s − 1.74·85-s + 0.530·95-s − 0.520·99-s − 1.28·101-s − 0.247·115-s − 2.88·119-s − 1.17·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.004273257276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004273257276\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 19 | $D_{4}$ | \( 1 + 6 p T + 74 p^{2} T^{2} + 6 p^{5} T^{3} + p^{8} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 23 T^{2} - 476 p^{2} T^{4} - 23 p^{8} T^{6} + p^{16} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 21 T + 1342 T^{2} + 21 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 68 T + 5301 T^{2} + 68 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 111 T + 27088 T^{2} + 111 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 77183 T^{2} + 3054117756 T^{4} - 77183 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 150 T + 162155 T^{2} - 150 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 39 T + 327028 T^{2} - 39 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1295951 T^{2} + 1154150796468 T^{4} - 1295951 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 348056 T^{2} + 1510187432718 T^{4} - 348056 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 3680824 T^{2} + 8018883544974 T^{4} - 3680824 p^{8} T^{6} + p^{16} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 4843928 T^{2} + 12077160453006 T^{4} - 4843928 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 1493 T + 5146446 T^{2} + 1493 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 3789 T + 13216636 T^{2} - 3789 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 28576543 T^{2} + 328095076878996 T^{4} - 28576543 p^{8} T^{6} + p^{16} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 20616935 T^{2} + 353317496460300 T^{4} - 20616935 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 79 T + 27159594 T^{2} - 79 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 69010271 T^{2} + 1992542074732308 T^{4} - 69010271 p^{8} T^{6} + p^{16} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 41313424 T^{2} + 12545129831538 p T^{4} - 41313424 p^{8} T^{6} + p^{16} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 7584 T + 69041153 T^{2} + 7584 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 1822376 p T^{2} + 8180923573778574 T^{4} - 1822376 p^{9} T^{6} + p^{16} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 16638 T + 162869650 T^{2} + 16638 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 52179052 T^{2} + 8356348223824470 T^{4} - 52179052 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 337141424 T^{2} + 44087261433798366 T^{4} - 337141424 p^{8} T^{6} + p^{16} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77395639644355840936388046810, −7.76959675882249401850853258762, −7.38153999516357350519600133896, −6.99011725459372157010407708849, −6.96567645820538918390800367044, −6.58050808331326829273379093625, −6.39968704896091100654125272360, −5.80887402305650930822460545202, −5.66855887360203556477159458117, −5.60113128705301450452892575227, −5.50298635050334125672515851099, −4.59114009077972828203548507805, −4.55684547014593037922451110945, −4.23977342745449628194473885405, −3.80516016835024293714567615105, −3.57345780723532813059484498856, −3.51839530639065244049410222790, −2.89807976016664205902446508767, −2.89515692724687354910718611722, −2.52324862913536761525375985235, −2.00312394551458528494248260999, −1.44772624061767177401290202639, −0.968792501342510269398009091666, −0.11761638518403575505452097930, −0.06281054938469694194333047115,
0.06281054938469694194333047115, 0.11761638518403575505452097930, 0.968792501342510269398009091666, 1.44772624061767177401290202639, 2.00312394551458528494248260999, 2.52324862913536761525375985235, 2.89515692724687354910718611722, 2.89807976016664205902446508767, 3.51839530639065244049410222790, 3.57345780723532813059484498856, 3.80516016835024293714567615105, 4.23977342745449628194473885405, 4.55684547014593037922451110945, 4.59114009077972828203548507805, 5.50298635050334125672515851099, 5.60113128705301450452892575227, 5.66855887360203556477159458117, 5.80887402305650930822460545202, 6.39968704896091100654125272360, 6.58050808331326829273379093625, 6.96567645820538918390800367044, 6.99011725459372157010407708849, 7.38153999516357350519600133896, 7.76959675882249401850853258762, 7.77395639644355840936388046810
