L(s) = 1 | + 0.618i·2-s + 0.618·3-s + 0.618·4-s + 0.381i·6-s + 1.61i·7-s + i·8-s − 0.618·9-s − i·11-s + 0.381·12-s − 1.00·14-s − 0.381i·18-s + 0.618i·19-s + 1.00i·21-s + 0.618·22-s + 1.61·23-s + 0.618i·24-s + ⋯ |
L(s) = 1 | + 0.618i·2-s + 0.618·3-s + 0.618·4-s + 0.381i·6-s + 1.61i·7-s + i·8-s − 0.618·9-s − i·11-s + 0.381·12-s − 1.00·14-s − 0.381i·18-s + 0.618i·19-s + 1.00i·21-s + 0.618·22-s + 1.61·23-s + 0.618i·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.586387097\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.586387097\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.618iT - T^{2} \) |
| 3 | \( 1 - 0.618T + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.61iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618iT - T^{2} \) |
| 23 | \( 1 - 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.61iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 0.618iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.61iT - T^{2} \) |
| 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
−9.095925369551746139538820557797, −8.814435850198006607853783526536, −8.137210326517093829113721998144, −7.38594170513729634414646024494, −6.27297140662719251114481611182, −5.74092703204870365036091661391, −5.19160015289070285605485097175, −3.48053015922686279154758778667, −2.75936600320791000665323816245, −1.97554491217688452470336438071,
1.14676397370918974089592620189, 2.32744956284831945345976316818, 3.21031578507652648803809475803, 3.99550743835623145647912398860, 4.90487023365986213254722904610, 6.25646414203093292681187990034, 7.14936399609368265597987068168, 7.45592176173952934808443526336, 8.437866300885635453894284120157, 9.560974235431865695038951133234