L(s) = 1 | + (−0.258 + 0.965i)3-s + (−0.258 − 0.965i)7-s + (−0.866 − 0.499i)9-s + 21-s + (−1.67 − 0.448i)23-s + (0.707 − 0.707i)27-s + (0.866 − 1.5i)29-s + (1.5 − 0.866i)41-s + (1.93 − 0.517i)43-s + (−1.67 + 0.448i)47-s + (0.5 − 0.866i)61-s + (−0.258 + 0.965i)63-s + (−0.965 − 0.258i)67-s + (0.866 − 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)3-s + (−0.258 − 0.965i)7-s + (−0.866 − 0.499i)9-s + 21-s + (−1.67 − 0.448i)23-s + (0.707 − 0.707i)27-s + (0.866 − 1.5i)29-s + (1.5 − 0.866i)41-s + (1.93 − 0.517i)43-s + (−1.67 + 0.448i)47-s + (0.5 − 0.866i)61-s + (−0.258 + 0.965i)63-s + (−0.965 − 0.258i)67-s + (0.866 − 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8798473813\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8798473813\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.93 + 0.517i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 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
−8.678960670127777763485867706822, −7.968889744970525982934578281206, −7.19300785358242256605268891080, −6.16317177064034551144489392738, −5.79823250573927423744052360142, −4.47963218903608088570211123846, −4.24120595708350516728273217507, −3.33421717189866293702980828726, −2.27001380822169051955014887185, −0.55543121089032741175334147648,
1.29257769535516837025109271187, 2.34162791252082398515028874401, 3.03045707631532064181963281224, 4.26026102505301081997255090441, 5.32380419763512549819421961055, 5.92264040112545291314511334881, 6.48877854302043528802135515553, 7.36363384274979480315403872988, 8.042858300943956755151844583529, 8.687510749690327559138348913124