L(s) = 1 | + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 + 0.831i)21-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + ⋯ |
L(s) = 1 | + (−0.555 − 0.831i)3-s + (0.195 + 0.980i)5-s + (−0.382 − 0.923i)7-s + (−0.382 + 0.923i)9-s + (0.831 + 0.555i)11-s + (0.195 − 0.980i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 + 0.831i)21-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (0.980 − 0.195i)27-s + (−0.831 + 0.555i)29-s − i·31-s − i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.248000552 - 0.4122836934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248000552 - 0.4122836934i\) |
\(L(1)\) |
\(\approx\) |
\(0.9223776759 - 0.1648705890i\) |
\(L(1)\) |
\(\approx\) |
\(0.9223776759 - 0.1648705890i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.555 - 0.831i)T \) |
| 5 | \( 1 + (0.195 + 0.980i)T \) |
| 7 | \( 1 + (-0.382 - 0.923i)T \) |
| 11 | \( 1 + (0.831 + 0.555i)T \) |
| 13 | \( 1 + (0.195 - 0.980i)T \) |
| 17 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (-0.923 - 0.382i)T \) |
| 29 | \( 1 + (-0.831 + 0.555i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.980 - 0.195i)T \) |
| 41 | \( 1 + (0.923 + 0.382i)T \) |
| 43 | \( 1 + (0.555 - 0.831i)T \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.831 - 0.555i)T \) |
| 59 | \( 1 + (-0.195 - 0.980i)T \) |
| 61 | \( 1 + (0.555 + 0.831i)T \) |
| 67 | \( 1 + (0.555 + 0.831i)T \) |
| 71 | \( 1 + (-0.382 - 0.923i)T \) |
| 73 | \( 1 + (0.382 - 0.923i)T \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.980 + 0.195i)T \) |
| 89 | \( 1 + (-0.923 + 0.382i)T \) |
| 97 | \( 1 - iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.67997436550156442136196320128, −18.84481873195707244016337542314, −18.10700289514035499470652688738, −17.20442957385689396567590577198, −16.646007998894530208622929399437, −16.04574246780576126518878464026, −15.67249269615142325024402647061, −14.51613506389797275887733638591, −14.01887968715151755322012380290, −12.96866224919100142966562292035, −12.07289704853770203085280587614, −11.79843876545134042199586188004, −11.03628606247768070916902741821, −9.807315455564616557680550533263, −9.37364262218377227232019083478, −8.93254979202230202694804145918, −8.03920550925816226114157632083, −6.75487064623685370226185240597, −5.842897387247823142248092565483, −5.6172006170637096804249268374, −4.53185021316094475207622159925, −3.95116169509561297330262602035, −2.99838734934929010085668430812, −1.80353520961063356929150750555, −0.76338114467335215474058018847,
0.676880745439637102383219903659, 1.651540809771103848864227598728, 2.55083018651162586518132154317, 3.56540255681667286207889814901, 4.25964026827585231739092534030, 5.62467491517258222705300022663, 6.14443281696394143372058472493, 6.81406507644464081456707370288, 7.61151846376545133633456163380, 7.96911178215999016097179964085, 9.422878694065687539470262316715, 10.14948533442978392599023224526, 10.78810943931568246430410084429, 11.359131837457763335991032920184, 12.3769404914258394403704950384, 12.87364227085724671469877096447, 13.67684604041073120371880184018, 14.39108775250247910308127829659, 14.91500579816583247994911985882, 16.03915332420627981263818106048, 16.80222637700285109930378133842, 17.45199423482058209331000748583, 17.900221475915580229993811244574, 18.743498061146835154142538676615, 19.30955169586194152883032534946