| L(s) = 1 | − i·2-s − 3-s − 4-s + i·6-s + 2i·7-s + i·8-s + 9-s + 12-s + (3 + 2i)13-s + 2·14-s + 16-s + 2·17-s − i·18-s − 6i·19-s − 2i·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.408i·6-s + 0.755i·7-s + 0.353i·8-s + 0.333·9-s + 0.288·12-s + (0.832 + 0.554i)13-s + 0.534·14-s + 0.250·16-s + 0.485·17-s − 0.235i·18-s − 1.37i·19-s − 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9248904348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9248904348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.233566324728389761112545598671, −8.872546661394092685504524929219, −7.82365178246178017393697956803, −6.86151622954446610530335012247, −5.95411416925638363048231088412, −5.28563608774151097138914480545, −4.36036393482824586285885686025, −3.42392193044851321230971858424, −2.32606893234971786995695680463, −1.23703861344529045297577011933,
0.39983921122863822593076142449, 1.76108113140118599472720261700, 3.67416244534569336265678746531, 4.00468394201600375013425422533, 5.38021803883807541705352987189, 5.80448407719749045938899759697, 6.63300166738852675882914303650, 7.58302181044842264236368034285, 7.986726649373977717944633570727, 8.956616267703219010374133862162
