L(s) = 1 | − 20·2-s + 120·4-s + 250·5-s − 100·7-s + 640·8-s − 5.00e3·10-s − 4.54e3·11-s + 3.54e3·13-s + 2.00e3·14-s − 1.15e4·16-s + 2.73e4·17-s + 3.87e4·19-s + 3.00e4·20-s + 9.08e4·22-s + 1.24e5·23-s + 4.68e4·25-s − 7.08e4·26-s − 1.20e4·28-s + 7.22e4·29-s + 3.06e5·31-s + 7.29e4·32-s − 5.46e5·34-s − 2.50e4·35-s − 1.23e5·37-s − 7.75e5·38-s + 1.60e5·40-s − 2.64e5·41-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 0.937·4-s + 0.894·5-s − 0.110·7-s + 0.441·8-s − 1.58·10-s − 1.02·11-s + 0.446·13-s + 0.194·14-s − 0.707·16-s + 1.34·17-s + 1.29·19-s + 0.838·20-s + 1.81·22-s + 2.12·23-s + 3/5·25-s − 0.789·26-s − 0.103·28-s + 0.550·29-s + 1.84·31-s + 0.393·32-s − 2.38·34-s − 0.0985·35-s − 0.399·37-s − 2.29·38-s + 0.395·40-s − 0.599·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.225124744\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.225124744\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + 5 p^{2} T + 35 p^{3} T^{2} + 5 p^{9} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 100 T + 1411250 T^{2} + 100 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4544 T + 31976326 T^{2} + 4544 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3540 T + 100535470 T^{2} - 3540 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 27340 T + 901005190 T^{2} - 27340 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2040 p T + 113449762 p T^{2} - 2040 p^{8} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 124140 T + 10649684530 T^{2} - 124140 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 72260 T + 6846819118 T^{2} - 72260 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 306824 T + 77964629966 T^{2} - 306824 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 123020 T + 144088599870 T^{2} + 123020 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 264364 T + 161786388886 T^{2} + 264364 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 423300 T + 446651231050 T^{2} - 423300 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 105460 T + 858715356610 T^{2} - 105460 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2391580 T + 3562552504510 T^{2} - 2391580 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1120120 T + 1362334883638 T^{2} - 1120120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2257044 T + 5613447576526 T^{2} - 2257044 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4516460 T + 16742087664890 T^{2} - 4516460 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 621784 T + 17914494152446 T^{2} + 621784 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4569060 T + 23424949855030 T^{2} - 4569060 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4333040 T + 330830231042 p T^{2} - 4333040 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9793020 T + 59971104320890 T^{2} - 9793020 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6025620 T + 89865866149558 T^{2} + 6025620 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4609540 T + 142930351581510 T^{2} - 4609540 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−14.59852360063796427733243724814, −14.00493737382901278011603018360, −13.39297133610143059644540145269, −13.03548884604674544987808746051, −12.15185124406984080145182921076, −11.45922302357553324708101202474, −10.45154286020594823189006120946, −10.37660375167141559171435437691, −9.507247035329806785202718061160, −9.416490350687852440185497505153, −8.369841317355816189714843271932, −8.163443991431038144273715191075, −7.27052843578853919982484826210, −6.53761132291264760759620629903, −5.37665655989887589678139222999, −5.04126837951953862533885090518, −3.37492276241368616820483871399, −2.50133437303367066785562798812, −0.970534731658867784971968568513, −0.896116664342765117196953890616,
0.896116664342765117196953890616, 0.970534731658867784971968568513, 2.50133437303367066785562798812, 3.37492276241368616820483871399, 5.04126837951953862533885090518, 5.37665655989887589678139222999, 6.53761132291264760759620629903, 7.27052843578853919982484826210, 8.163443991431038144273715191075, 8.369841317355816189714843271932, 9.416490350687852440185497505153, 9.507247035329806785202718061160, 10.37660375167141559171435437691, 10.45154286020594823189006120946, 11.45922302357553324708101202474, 12.15185124406984080145182921076, 13.03548884604674544987808746051, 13.39297133610143059644540145269, 14.00493737382901278011603018360, 14.59852360063796427733243724814