L(s) = 1 | − 9-s + 2i·11-s + 2·17-s + i·19-s + 25-s − 2i·43-s − 49-s − 2·73-s + 81-s + 2i·83-s − 2i·99-s + ⋯ |
L(s) = 1 | − 9-s + 2i·11-s + 2·17-s + i·19-s + 25-s − 2i·43-s − 49-s − 2·73-s + 81-s + 2i·83-s − 2i·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.004877544\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004877544\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 2iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 2T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 2iT - T^{2} \) |
| 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
−10.03858683488733557500576328186, −9.336471726877595065706323718001, −8.318244500592858537973108643467, −7.60113094711622986676220564521, −6.83392596893603686582455254946, −5.69307302927373193768143545120, −5.06060394405552271932057231691, −3.90882605367159703739915161423, −2.86225146813457197621463621780, −1.62813312193582553836094851059,
0.973135725108569989388929430460, 2.98582072277298909962376996114, 3.27243948113723452794469111396, 4.84572381246513044076514900335, 5.74416293752405322785256604644, 6.25276101971310853271943309746, 7.51386320559291969345131698271, 8.332150890540909515735825309940, 8.846577655230746483293077608280, 9.788327927997988828700733405486