| L(s) = 1 | − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s + 11-s − 2·13-s − 14-s − 16-s + 6·17-s + 4·19-s + 20-s − 22-s + 4·23-s + 25-s + 2·26-s − 28-s + 2·29-s − 4·31-s − 5·32-s − 6·34-s − 35-s − 2·37-s − 4·38-s − 3·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.718·31-s − 0.883·32-s − 1.02·34-s − 0.169·35-s − 0.328·37-s − 0.648·38-s − 0.474·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.054378855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054378855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.601407343299504331187533634349, −7.78349983307565944610757287206, −7.51903221037529297623393965152, −6.56023779437574610968134261995, −5.24790103860362103170152473563, −4.98257720003110949062912836057, −3.85943577306988298897974594828, −3.12443953727595584293512581995, −1.65258110320021470193428110339, −0.72417743423215125020506960714,
0.72417743423215125020506960714, 1.65258110320021470193428110339, 3.12443953727595584293512581995, 3.85943577306988298897974594828, 4.98257720003110949062912836057, 5.24790103860362103170152473563, 6.56023779437574610968134261995, 7.51903221037529297623393965152, 7.78349983307565944610757287206, 8.601407343299504331187533634349
