| L(s) = 1 | − 2-s + 4-s − 0.618·7-s − 8-s + 9-s − 1.61·11-s + 1.61·13-s + 0.618·14-s + 16-s − 18-s + 0.618·19-s + 1.61·22-s + 1.61·23-s − 1.61·26-s − 0.618·28-s − 32-s + 36-s − 0.618·37-s − 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s + 1.61·47-s − 0.618·49-s + 1.61·52-s − 0.618·53-s + 0.618·56-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 0.618·7-s − 8-s + 9-s − 1.61·11-s + 1.61·13-s + 0.618·14-s + 16-s − 18-s + 0.618·19-s + 1.61·22-s + 1.61·23-s − 1.61·26-s − 0.618·28-s − 32-s + 36-s − 0.618·37-s − 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s + 1.61·47-s − 0.618·49-s + 1.61·52-s − 0.618·53-s + 0.618·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6767991991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6767991991\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.618T + T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 - 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618T + T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.61T + T^{2} \) |
| 53 | \( 1 + 0.618T + T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 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
−10.21909375937087143863763296476, −9.336281646077878582184375078988, −8.606313541949706043374262076575, −7.68557501242738725718451301627, −7.03934487728620344713907360437, −6.12132479717912161917720685954, −5.15686831524762743437841247795, −3.62933963483017492191714838091, −2.67523889781754086218996491939, −1.18834119397106713934378917459,
1.18834119397106713934378917459, 2.67523889781754086218996491939, 3.62933963483017492191714838091, 5.15686831524762743437841247795, 6.12132479717912161917720685954, 7.03934487728620344713907360437, 7.68557501242738725718451301627, 8.606313541949706043374262076575, 9.336281646077878582184375078988, 10.21909375937087143863763296476
