L(s) = 1 | + (−0.866 − 0.5i)5-s + (1.36 + 0.366i)7-s + (0.5 + 0.866i)11-s + (−1 + i)17-s − i·19-s + (1.36 − 0.366i)23-s + (0.499 + 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (−0.999 − i)35-s + (0.5 − 0.866i)41-s + (0.866 + 0.5i)49-s + (1 + i)53-s − 0.999i·55-s + (−0.866 − 0.5i)59-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)5-s + (1.36 + 0.366i)7-s + (0.5 + 0.866i)11-s + (−1 + i)17-s − i·19-s + (1.36 − 0.366i)23-s + (0.499 + 0.866i)25-s + (0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (−0.999 − i)35-s + (0.5 − 0.866i)41-s + (0.866 + 0.5i)49-s + (1 + i)53-s − 0.999i·55-s + (−0.866 − 0.5i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.142787008\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.142787008\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
good | 7 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T^{2} \) |
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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
−9.257483636899931633157939478719, −8.787342331340225730153368429425, −8.128900819292203494607398730448, −7.27860524952347578626669165667, −6.56114341550822524922673303922, −5.16245144681585178975938557281, −4.67501408808301547431104228098, −3.94237527514091366412939739054, −2.47011711810938042525988587645, −1.33007660637035564411500531200,
1.15008247787606334460081597165, 2.63982610821641598504701724611, 3.71159123079632548022904529547, 4.50670400782096265321526619677, 5.33612847460532026130606893651, 6.53419131199815486750866616118, 7.24764986247075998111312739145, 8.006629972995939352247845306014, 8.591212676540333590995160135584, 9.463406605176236402091612393858