L(s) = 1 | + 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 5-s + 2.00·6-s − 7-s − 1.00·9-s − 1.41i·10-s − 11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s + 1.41i·15-s − 0.999·16-s + 1.41i·17-s − 1.41i·18-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 5-s + 2.00·6-s − 7-s − 1.00·9-s − 1.41i·10-s − 11-s + 1.41i·12-s + 1.41i·13-s − 1.41i·14-s + 1.41i·15-s − 0.999·16-s + 1.41i·17-s − 1.41i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1099 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2979770427\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2979770427\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 - 1.41iT - T^{2} \) |
| 3 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.46829192724862959251865565560, −9.050300994705827572855263228799, −8.542538496249118108904771575075, −7.52162000766030121997088979968, −7.24243998820386445480635695261, −6.56221271945391349652557603056, −5.82888592592559224138222492326, −4.68319198866827924097166800081, −3.42465057177099296315430515807, −1.98831139872637300240300375714,
0.25379040125094437141265217263, 2.74764581226365098919858753969, 3.24586272782000393369981144265, 4.05683894913961140945146289398, 4.84992475220331956081163210841, 5.93298931425247340172289471092, 7.39968368005523691050150253004, 8.263320345692457528628780668144, 9.317730123173970563400941452994, 10.02697127879465102286653247363