L(s) = 1 | − 2-s − 1.92·3-s + 4-s + 0.200·5-s + 1.92·6-s − 3.44·7-s − 8-s + 0.708·9-s − 0.200·10-s + 11-s − 1.92·12-s − 1.33·13-s + 3.44·14-s − 0.386·15-s + 16-s − 5.38·17-s − 0.708·18-s + 2.98·19-s + 0.200·20-s + 6.62·21-s − 22-s + 2.46·23-s + 1.92·24-s − 4.95·25-s + 1.33·26-s + 4.41·27-s − 3.44·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.11·3-s + 0.5·4-s + 0.0897·5-s + 0.786·6-s − 1.30·7-s − 0.353·8-s + 0.236·9-s − 0.0634·10-s + 0.301·11-s − 0.555·12-s − 0.371·13-s + 0.920·14-s − 0.0997·15-s + 0.250·16-s − 1.30·17-s − 0.167·18-s + 0.684·19-s + 0.0448·20-s + 1.44·21-s − 0.213·22-s + 0.513·23-s + 0.393·24-s − 0.991·25-s + 0.262·26-s + 0.849·27-s − 0.650·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4462686764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4462686764\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 1.92T + 3T^{2} \) |
| 5 | \( 1 - 0.200T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 13 | \( 1 + 1.33T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 - 4.75T + 41T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 - 0.276T + 61T^{2} \) |
| 67 | \( 1 - 5.63T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 + 7.88T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 7.15T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−9.970415366553617965212416396654, −9.416518394845703661580270347735, −8.546992192007234941987843534128, −7.29216883805270581143470293973, −6.58526669208972412617156424213, −6.00218782662950707501932570031, −5.00130671703935781971430619756, −3.64422878164724830179059992733, −2.38296095564228870083324786349, −0.58846266957470291786098093633,
0.58846266957470291786098093633, 2.38296095564228870083324786349, 3.64422878164724830179059992733, 5.00130671703935781971430619756, 6.00218782662950707501932570031, 6.58526669208972412617156424213, 7.29216883805270581143470293973, 8.546992192007234941987843534128, 9.416518394845703661580270347735, 9.970415366553617965212416396654