| L(s) = 1 | − 8·2-s + 40·4-s − 160·8-s − 18·9-s + 74·13-s + 560·16-s + 30·17-s + 144·18-s − 154·19-s + 175·25-s − 592·26-s − 1.79e3·32-s − 240·34-s − 720·36-s + 1.23e3·38-s − 70·43-s + 396·47-s + 1.30e3·49-s − 1.40e3·50-s + 2.96e3·52-s − 1.68e3·53-s − 480·59-s + 5.37e3·64-s + 248·67-s + 1.20e3·68-s + 2.88e3·72-s − 6.16e3·76-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s − 2/3·9-s + 1.57·13-s + 35/4·16-s + 0.428·17-s + 1.88·18-s − 1.85·19-s + 7/5·25-s − 4.46·26-s − 9.89·32-s − 1.21·34-s − 3.33·36-s + 5.25·38-s − 0.248·43-s + 1.22·47-s + 3.79·49-s − 3.95·50-s + 7.89·52-s − 4.35·53-s − 1.05·59-s + 21/2·64-s + 0.452·67-s + 2.14·68-s + 4.71·72-s − 9.29·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6361713254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6361713254\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 30 T + 290 p T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 7 p^{2} T^{2} + 27384 T^{4} - 7 p^{8} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 650 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 425 p T^{2} + 8903328 T^{4} - 425 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 37 T + 3456 T^{2} - 37 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 77 T + 3678 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 29215 T^{2} + 432603828 T^{4} - 29215 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 92716 T^{2} + 3337399830 T^{4} - 92716 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 58682 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 143212 T^{2} + 9503749878 T^{4} - 143212 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 219403 T^{2} + 20944379844 T^{4} - 219403 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 35 T + 4410 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 198 T + 89422 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 840 T + 453670 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{4} \) |
| 61 | $D_4\times C_2$ | \( 1 - 257548 T^{2} + 94657264854 T^{4} - 257548 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 124 T + 584886 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 492568 T^{2} + 206508228654 T^{4} - 492568 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 368608 T^{2} + 39756704478 T^{4} - 368608 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 788584 T^{2} + 404149084206 T^{4} - 788584 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1770 T + 1880710 T^{2} - 1770 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 444 T + 1131478 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2450020 T^{2} + 2996691020358 T^{4} - 2450020 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−9.695966033086685430420549627491, −9.218184145887327907738600030728, −9.042521378452423631273749774085, −8.870610500223023139850223747211, −8.657705187848134796837600979211, −8.372885957577670235863810315341, −8.041536942676731091916432450972, −7.86487536368302905038387328748, −7.50424497851224817326838947746, −7.10194265539953226348158025592, −6.88135934617782915271137837108, −6.41551338583994392128849918063, −6.10958228119717253897783093867, −6.07439556634535375665088669931, −5.70291634479394383871040728173, −4.99296197028923209166817952310, −4.64006429441394461990934876903, −3.99669118894890507799124540465, −3.30814031622808784755732203491, −3.30645690331890114178229099600, −2.46808041724115374360011231755, −2.16168740136903158135001125787, −1.56421717156336396739706385155, −0.894268914611103023861360536446, −0.48253064571467723077177950324,
0.48253064571467723077177950324, 0.894268914611103023861360536446, 1.56421717156336396739706385155, 2.16168740136903158135001125787, 2.46808041724115374360011231755, 3.30645690331890114178229099600, 3.30814031622808784755732203491, 3.99669118894890507799124540465, 4.64006429441394461990934876903, 4.99296197028923209166817952310, 5.70291634479394383871040728173, 6.07439556634535375665088669931, 6.10958228119717253897783093867, 6.41551338583994392128849918063, 6.88135934617782915271137837108, 7.10194265539953226348158025592, 7.50424497851224817326838947746, 7.86487536368302905038387328748, 8.041536942676731091916432450972, 8.372885957577670235863810315341, 8.657705187848134796837600979211, 8.870610500223023139850223747211, 9.042521378452423631273749774085, 9.218184145887327907738600030728, 9.695966033086685430420549627491
