L(s) = 1 | + 1.73·3-s − 5.19·7-s + 2.99·9-s + 7i·13-s + 5.19i·19-s − 9·21-s + 5.19·27-s + 1.73i·31-s + 10i·37-s + 12.1i·39-s − 1.73·43-s + 20·49-s + 9i·57-s − 61-s − 15.5·63-s + ⋯ |
L(s) = 1 | + 1.00·3-s − 1.96·7-s + 0.999·9-s + 1.94i·13-s + 1.19i·19-s − 1.96·21-s + 1.00·27-s + 0.311i·31-s + 1.64i·37-s + 1.94i·39-s − 0.264·43-s + 2.85·49-s + 1.19i·57-s − 0.128·61-s − 1.96·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.483583614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483583614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 5.19T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
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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
−9.843954975854656828477445957845, −9.174075088356045731340388198781, −8.570845947295107052848144113078, −7.39017096557644175614159673508, −6.69477016761271122260977606262, −6.09219501933549044042707567337, −4.48921633860758638676170169738, −3.67017533112789168414992746728, −2.91343342706374562342221559431, −1.69190996967345702015385543830,
0.54396303043683315856031446211, 2.57865312731521128586671110581, 3.13719144071411459267368978091, 3.96508712736863941057538495972, 5.35585424263450399598588658806, 6.31518126145457791205752855917, 7.15893934530357559403133607722, 7.87007210639657550789323976059, 8.922581265018907633205072868441, 9.428233673211750949047553252363