L(s) = 1 | + (0.928 + 0.371i)2-s + (0.999 + 0.0237i)3-s + (0.723 + 0.690i)4-s + (0.909 − 0.415i)5-s + (0.919 + 0.393i)6-s + (0.986 − 0.165i)7-s + (0.415 + 0.909i)8-s + (0.998 + 0.0475i)9-s + (0.998 − 0.0475i)10-s + (−0.580 − 0.814i)11-s + (0.707 + 0.707i)12-s + (0.977 + 0.212i)14-s + (0.919 − 0.393i)15-s + (0.0475 + 0.998i)16-s + (−0.371 − 0.928i)17-s + (0.909 + 0.415i)18-s + ⋯ |
L(s) = 1 | + (0.928 + 0.371i)2-s + (0.999 + 0.0237i)3-s + (0.723 + 0.690i)4-s + (0.909 − 0.415i)5-s + (0.919 + 0.393i)6-s + (0.986 − 0.165i)7-s + (0.415 + 0.909i)8-s + (0.998 + 0.0475i)9-s + (0.998 − 0.0475i)10-s + (−0.580 − 0.814i)11-s + (0.707 + 0.707i)12-s + (0.977 + 0.212i)14-s + (0.919 − 0.393i)15-s + (0.0475 + 0.998i)16-s + (−0.371 − 0.928i)17-s + (0.909 + 0.415i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(8.853571593 + 0.7967514582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.853571593 + 0.7967514582i\) |
\(L(1)\) |
\(\approx\) |
\(3.425059825 + 0.4196580526i\) |
\(L(1)\) |
\(\approx\) |
\(3.425059825 + 0.4196580526i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (0.928 + 0.371i)T \) |
| 3 | \( 1 + (0.999 + 0.0237i)T \) |
| 5 | \( 1 + (0.909 - 0.415i)T \) |
| 7 | \( 1 + (0.986 - 0.165i)T \) |
| 11 | \( 1 + (-0.580 - 0.814i)T \) |
| 17 | \( 1 + (-0.371 - 0.928i)T \) |
| 19 | \( 1 + (0.672 + 0.739i)T \) |
| 23 | \( 1 + (-0.952 - 0.304i)T \) |
| 29 | \( 1 + (0.771 - 0.636i)T \) |
| 31 | \( 1 + (0.212 - 0.977i)T \) |
| 37 | \( 1 + (-0.258 - 0.965i)T \) |
| 41 | \( 1 + (-0.853 + 0.520i)T \) |
| 43 | \( 1 + (0.771 + 0.636i)T \) |
| 47 | \( 1 + (0.281 + 0.959i)T \) |
| 53 | \( 1 + (-0.281 + 0.959i)T \) |
| 59 | \( 1 + (-0.999 + 0.0237i)T \) |
| 61 | \( 1 + (0.899 - 0.436i)T \) |
| 67 | \( 1 + (0.723 - 0.690i)T \) |
| 71 | \( 1 + (0.0950 + 0.995i)T \) |
| 73 | \( 1 + (-0.841 + 0.540i)T \) |
| 79 | \( 1 + (0.540 + 0.841i)T \) |
| 83 | \( 1 + (0.800 - 0.599i)T \) |
| 97 | \( 1 + (0.580 - 0.814i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.04756327492907949136256304014, −20.471067192360328442116347004735, −19.84257669019612903199105634917, −18.88695441328256767079029501959, −18.04214901516055320039593524707, −17.502207142753460122569277525186, −15.98823592028034726595885911725, −15.2358601860157698247414235412, −14.74587034447361773325446578042, −13.8961070601748169544593273326, −13.551998918037739576901766918078, −12.603397888741330213079750465181, −11.83274792309365451560528941976, −10.55354446576862325491734699274, −10.28138633915756323734195613213, −9.2483458287846754924461433692, −8.258618214673128728849189250480, −7.24465647731689958126747270833, −6.56377827239796488706241339734, −5.32017062406873115293349582185, −4.75953881766219190702654955208, −3.677178237646204732338075209348, −2.69982247300167610538512156928, −1.99613747197766507129225332168, −1.40015645118779623982499146157,
1.12083333205327107618229905319, 2.20161884892136794953446672924, 2.781404101230192669500009924117, 3.98783973113224276840044514820, 4.75379998354762671788115381746, 5.566757914069796300706773639941, 6.433852087091618017029764323446, 7.69549231321871123316981800336, 8.05253629506759908136327132232, 8.98621238204019997217767340283, 9.984033509635454840567423066067, 10.91499164835957699833833196203, 11.90197686005172967110841022276, 12.8014969030353140389255102714, 13.68019428087088614590873832647, 13.98508115125277727397908638600, 14.534843252884247285109709144133, 15.698196982133443989026313564167, 16.13401974699585933298472372265, 17.142803942164268098744398856940, 17.999352179828566231566079467875, 18.719352716030734228859066430495, 20.094542157821538366052990233192, 20.51008076682337501208780398472, 21.1571818348231353860309727285