L(s) = 1 | + i·3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s − 9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)12-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + i·17-s + (−0.5 − 0.866i)21-s − i·27-s + 0.999i·28-s + (1 + 1.73i)29-s + (−0.866 + 0.5i)33-s + (−0.5 + 0.866i)36-s + ⋯ |
L(s) = 1 | + i·3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s − 9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)12-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + i·17-s + (−0.5 − 0.866i)21-s − i·27-s + 0.999i·28-s + (1 + 1.73i)29-s + (−0.866 + 0.5i)33-s + (−0.5 + 0.866i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.120891360\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120891360\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
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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
−9.882945718684564765977394015568, −9.061509463514713855699468696296, −8.612159120527712286954596167642, −7.10948742765754510011403489520, −6.36043914413746361929217544352, −5.77847644038558579992083573489, −4.82148606476732153881546481254, −3.88473803549106906982580107473, −2.89592097041120463039240858315, −1.65675978560010566305981127842,
0.933100740722115110091882544548, 2.57117304024433075536279854819, 3.22163759071610718594362779710, 4.14384563040902869233923008956, 5.79071754921210242733575081470, 6.37228678742432010156662628566, 7.04617732380829806092168981110, 7.81416484697865207993431669241, 8.476150256186513029777450972853, 9.234390226051045123538238514807