| L(s) = 1 | − 2-s + 4-s − 0.618·5-s − 8-s + 0.618·10-s − 0.618·11-s − 1.61·13-s + 16-s − 0.618·20-s + 0.618·22-s + 1.61·23-s − 0.618·25-s + 1.61·26-s + 1.61·29-s + 0.618·31-s − 32-s + 37-s + 0.618·40-s + 1.61·41-s − 0.618·44-s − 1.61·46-s + 49-s + 0.618·50-s − 1.61·52-s + 0.381·55-s − 1.61·58-s + 0.618·61-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 0.618·5-s − 8-s + 0.618·10-s − 0.618·11-s − 1.61·13-s + 16-s − 0.618·20-s + 0.618·22-s + 1.61·23-s − 0.618·25-s + 1.61·26-s + 1.61·29-s + 0.618·31-s − 32-s + 37-s + 0.618·40-s + 1.61·41-s − 0.618·44-s − 1.61·46-s + 49-s + 0.618·50-s − 1.61·52-s + 0.381·55-s − 1.61·58-s + 0.618·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6015590468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6015590468\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 0.618T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.618T + T^{2} \) |
| 13 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.61T + T^{2} \) |
| 29 | \( 1 - 1.61T + T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 - 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 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
−8.979374712579495712556158665599, −8.274634844347233882051705482470, −7.50120339307721629264083526537, −7.16418426001485927859536252927, −6.16502955346187186860172135688, −5.17038634608667713165506486556, −4.32457938562147200984287424066, −2.94296595494778320223885697617, −2.43173642564324971593775020106, −0.811964214674075218398817175069,
0.811964214674075218398817175069, 2.43173642564324971593775020106, 2.94296595494778320223885697617, 4.32457938562147200984287424066, 5.17038634608667713165506486556, 6.16502955346187186860172135688, 7.16418426001485927859536252927, 7.50120339307721629264083526537, 8.274634844347233882051705482470, 8.979374712579495712556158665599
