| L(s) = 1 | − i·2-s − 3-s − 4-s + 2i·5-s + i·6-s + i·8-s + 9-s + 2·10-s + 12-s + (−2 + 3i)13-s − 2i·15-s + 16-s + 2·17-s − i·18-s + 4i·19-s − 2i·20-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.894i·5-s + 0.408i·6-s + 0.353i·8-s + 0.333·9-s + 0.632·10-s + 0.288·12-s + (−0.554 + 0.832i)13-s − 0.516i·15-s + 0.250·16-s + 0.485·17-s − 0.235i·18-s + 0.917i·19-s − 0.447i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3913804180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3913804180\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (2 - 3i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.998310390228690436984839026373, −7.930285637752151492286165146755, −7.39015759565747129628504855459, −6.42387833340555880189861123424, −5.92106320490358142633258051317, −4.89960585548359665993483001124, −4.13427384757802814094378207669, −3.31645978133578997846495246499, −2.36445560711683178088318063611, −1.43978161865337530302982254744,
0.13835932908597071675205284041, 1.14569926967265021163286808671, 2.57569096812133945889563895155, 3.84849602046665846154564978220, 4.62996282583798631118008252949, 5.33462430195331654459716392519, 5.78026219505288544744416246735, 6.73034273168710720939630488159, 7.45266808206281926041915776106, 8.122355185600567942613201423498
