L(s) = 1 | − 3-s + (2 + i)5-s + 4i·7-s + 9-s + 4·13-s + (−2 − i)15-s + 8i·19-s − 4i·21-s + 4i·23-s + (3 + 4i)25-s − 27-s − 6i·29-s + 8·31-s + (−4 + 8i)35-s − 4·37-s + ⋯ |
L(s) = 1 | − 0.577·3-s + (0.894 + 0.447i)5-s + 1.51i·7-s + 0.333·9-s + 1.10·13-s + (−0.516 − 0.258i)15-s + 1.83i·19-s − 0.872i·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 0.192·27-s − 1.11i·29-s + 1.43·31-s + (−0.676 + 1.35i)35-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.907093476\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907093476\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-2 - i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.643341868131511253919080652131, −8.174774246779822126799057410626, −7.03867932419971488116179221426, −6.19659691040763031589308609790, −5.76603369735828493651123605100, −5.39294326186920822235473980826, −4.11529057434854098221539106148, −3.15912533813683896638130809239, −2.18095538367954224802715980440, −1.38670433189134003176200789978,
0.66490024439535186336475027711, 1.30946335159695553694491586273, 2.63868692538247275934287778627, 3.77837432412368337841722462532, 4.62351507821192938027965605164, 5.12563768601513364714137869726, 6.19449141239918099876392670374, 6.71883504358325432517094108402, 7.28467451916425409737504924828, 8.439280947041724828369778986336