L(s) = 1 | + i·2-s + i·3-s − 4-s + 0.942i·5-s − 6-s − i·8-s − 9-s − 0.942·10-s + (1.67 + 2.86i)11-s − i·12-s + 2.81·13-s − 0.942·15-s + 16-s + 4.30·17-s − i·18-s + 2.51·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.421i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.298·10-s + (0.503 + 0.863i)11-s − 0.288i·12-s + 0.781·13-s − 0.243·15-s + 0.250·16-s + 1.04·17-s − 0.235i·18-s + 0.577·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.092086683\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.092086683\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-1.67 - 2.86i)T \) |
good | 5 | \( 1 - 0.942iT - 5T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 - 4.33T + 23T^{2} \) |
| 29 | \( 1 + 8.01iT - 29T^{2} \) |
| 31 | \( 1 + 6.05iT - 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 8.96iT - 47T^{2} \) |
| 53 | \( 1 - 4.55T + 53T^{2} \) |
| 59 | \( 1 - 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 2.41T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 + 9.26iT - 79T^{2} \) |
| 83 | \( 1 - 8.84T + 83T^{2} \) |
| 89 | \( 1 - 11.5iT - 89T^{2} \) |
| 97 | \( 1 - 12.5iT - 97T^{2} \) |
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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
−8.912058992828403230653157273890, −8.003802552390160973982444552922, −7.36161959549522665907152613317, −6.60938386992105077005294646370, −5.84110942704316858610112324141, −5.13489027722733000179783258108, −4.19635651971068278209289869613, −3.57879129555047220015292767590, −2.48225116030831156224882859696, −0.948432344752876277217613410137,
1.02244794552740181983315126933, 1.35326956361573811765407003149, 3.03365199464561167679597070018, 3.31075373234622980042441101732, 4.58884486524032069267925932641, 5.35629960363447668051717825354, 6.14885792762905833791441129437, 6.95325130610505259886680991258, 7.922830391815377907599350085860, 8.542766345891091491759135740778