L(s) = 1 | − 2-s − 3-s + 4-s + 2.56·5-s + 6-s + 3.96·7-s − 8-s + 9-s − 2.56·10-s − 3.64·11-s − 12-s − 13-s − 3.96·14-s − 2.56·15-s + 16-s + 6.34·17-s − 18-s + 6.47·19-s + 2.56·20-s − 3.96·21-s + 3.64·22-s + 5.03·23-s + 24-s + 1.60·25-s + 26-s − 27-s + 3.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.14·5-s + 0.408·6-s + 1.49·7-s − 0.353·8-s + 0.333·9-s − 0.812·10-s − 1.10·11-s − 0.288·12-s − 0.277·13-s − 1.05·14-s − 0.663·15-s + 0.250·16-s + 1.53·17-s − 0.235·18-s + 1.48·19-s + 0.574·20-s − 0.864·21-s + 0.778·22-s + 1.04·23-s + 0.204·24-s + 0.320·25-s + 0.196·26-s − 0.192·27-s + 0.748·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.139401322\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.139401322\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 - 5.03T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 8.81T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 - 0.106T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 2.03T + 67T^{2} \) |
| 71 | \( 1 - 0.802T + 71T^{2} \) |
| 73 | \( 1 - 8.44T + 73T^{2} \) |
| 79 | \( 1 - 0.928T + 79T^{2} \) |
| 83 | \( 1 + 6.12T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 9.59T + 97T^{2} \) |
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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
−7.69821073350271609287203610106, −7.46371927831500523844891278164, −6.39982043151793176824265656020, −5.66443664231051451794299637964, −5.13365155288714455941813030703, −4.75897381671482587437038701851, −3.19709326198635350182080402740, −2.47279350519668054954130853325, −1.46444779705088097774795430065, −0.945936293955441338155437069543,
0.945936293955441338155437069543, 1.46444779705088097774795430065, 2.47279350519668054954130853325, 3.19709326198635350182080402740, 4.75897381671482587437038701851, 5.13365155288714455941813030703, 5.66443664231051451794299637964, 6.39982043151793176824265656020, 7.46371927831500523844891278164, 7.69821073350271609287203610106