L(s) = 1 | − 2-s + 0.347·3-s + 4-s − 5-s − 0.347·6-s − 8-s − 2.87·9-s + 10-s − 11-s + 0.347·12-s − 6.45·13-s − 0.347·15-s + 16-s − 2.12·17-s + 2.87·18-s + 5.90·19-s − 20-s + 22-s − 8.58·23-s − 0.347·24-s + 25-s + 6.45·26-s − 2.04·27-s − 9.66·29-s + 0.347·30-s − 8.35·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.200·3-s + 0.5·4-s − 0.447·5-s − 0.141·6-s − 0.353·8-s − 0.959·9-s + 0.316·10-s − 0.301·11-s + 0.100·12-s − 1.78·13-s − 0.0896·15-s + 0.250·16-s − 0.514·17-s + 0.678·18-s + 1.35·19-s − 0.223·20-s + 0.213·22-s − 1.78·23-s − 0.0708·24-s + 0.200·25-s + 1.26·26-s − 0.392·27-s − 1.79·29-s + 0.0634·30-s − 1.50·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5071060774\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5071060774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 13 | \( 1 + 6.45T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 + 8.58T + 23T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 - 3.51T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 - 0.389T + 73T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 9.29T + 89T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
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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
−7.963216260676677579723784652138, −7.60098217686892199771428213518, −7.17209857170854968979685911457, −5.88657240230911053075473457829, −5.52947675922019981913683579447, −4.45552772870677711305824760448, −3.54768975021356284728597272250, −2.61086464320155678627574999297, −2.02107807957531068283684410538, −0.39204086576521622300986012386,
0.39204086576521622300986012386, 2.02107807957531068283684410538, 2.61086464320155678627574999297, 3.54768975021356284728597272250, 4.45552772870677711305824760448, 5.52947675922019981913683579447, 5.88657240230911053075473457829, 7.17209857170854968979685911457, 7.60098217686892199771428213518, 7.963216260676677579723784652138