L(s) = 1 | + 1.41i·5-s + 2i·7-s − 1.41·11-s − 1.41i·17-s + i·19-s + 1.41·23-s − 1.00·25-s − 2.82·35-s − 1.41·47-s − 3·49-s − 2.00i·55-s + 2·73-s − 2.82i·77-s − 1.41·83-s + 2.00·85-s + ⋯ |
L(s) = 1 | + 1.41i·5-s + 2i·7-s − 1.41·11-s − 1.41i·17-s + i·19-s + 1.41·23-s − 1.00·25-s − 2.82·35-s − 1.41·47-s − 3·49-s − 2.00i·55-s + 2·73-s − 2.82i·77-s − 1.41·83-s + 2.00·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9469997819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9469997819\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 - 2iT - T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.400173048581408354543022018356, −8.496132847076799336790071366200, −7.81849461862177045661346444372, −7.00231306528612849716364554609, −6.25217281058969995526214465318, −5.42344790612775697933073101832, −4.96264673909214643984263046458, −3.23332636057064379034778511041, −2.81507980698047955357329210594, −2.12337318778334656757748351126,
0.59956485099503940508533282192, 1.58693355816893826856191716342, 3.11381557447035940134530015061, 4.11610068772940254664088502274, 4.76650149561649759856095958143, 5.32054269633031846868438154301, 6.54478891107410618145235976902, 7.29871827647541891227300543365, 8.032131492031980649612570800033, 8.525205382485707880562984109575