L(s) = 1 | − i·2-s − 4-s + 1.73·5-s + i·8-s − 1.73i·10-s + i·11-s + 16-s − 1.73·20-s + 22-s + 1.99·25-s − i·29-s + 1.73i·31-s − i·32-s + 1.73i·40-s − i·44-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + 1.73·5-s + i·8-s − 1.73i·10-s + i·11-s + 16-s − 1.73·20-s + 22-s + 1.99·25-s − i·29-s + 1.73i·31-s − i·32-s + 1.73i·40-s − i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.587741419\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.587741419\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.73T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 - 1.73T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−8.951939027419018178763938646644, −8.199718815188268912435577511211, −7.06948644922288458426827026949, −6.30652242895928354825872792437, −5.38057765933395831952489489160, −4.93600730079001543264896680910, −3.91707084961185202064818833665, −2.77191738708926786950871093926, −2.10249549315054070032698643319, −1.33338235996394542513060076790,
1.11047489824872060912961662568, 2.36811570449277919607665540668, 3.43243593251433773561261784100, 4.52528228267829412966910626289, 5.48979427799134967975763327924, 5.80513005592727699718192273727, 6.48069265154468218136253417991, 7.21201317792271912714305854405, 8.176826574185748723248119962229, 8.855367932406073764821197033003