L(s) = 1 | − 0.517i·2-s + 0.732·4-s − 0.896i·8-s + 1.93i·11-s + 0.267·16-s + 0.999·22-s + 1.41i·23-s + 25-s + 1.41i·29-s − 1.03i·32-s − 1.73·37-s + 1.73·43-s + 1.41i·44-s + 0.732·46-s − 0.517i·50-s + ⋯ |
L(s) = 1 | − 0.517i·2-s + 0.732·4-s − 0.896i·8-s + 1.93i·11-s + 0.267·16-s + 0.999·22-s + 1.41i·23-s + 25-s + 1.41i·29-s − 1.03i·32-s − 1.73·37-s + 1.73·43-s + 1.41i·44-s + 0.732·46-s − 0.517i·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.616730671\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.616730671\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.517iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.93iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.73T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.93iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 + 0.517iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−8.764944194589420876815094318355, −7.68458949971094675799795762696, −7.05538804049269038003967621563, −6.77383929399435320026884319362, −5.55704033591448449378196093489, −4.86811669000795365775407743210, −3.88407554631482228822492648406, −3.09662523196564915871590242491, −2.08183176202678648937278717595, −1.44785938348991474911451193446,
0.951767437125475136559348898996, 2.40539446366987347553481715892, 3.03253884671891232351926797042, 4.04203104024629401648405164716, 5.12433118170940068414114745194, 5.92555473152344411233792404851, 6.30889748297332492022257215931, 7.12244528102868276957012557363, 7.901876518305277355443048978798, 8.574668238904785647650521647541