| L(s) = 1 | − 2-s − 3-s + 4-s + 2.51·5-s + 6-s + 4.42·7-s − 8-s + 9-s − 2.51·10-s + 1.04·11-s − 12-s − 0.0694·13-s − 4.42·14-s − 2.51·15-s + 16-s − 17-s − 18-s + 0.121·19-s + 2.51·20-s − 4.42·21-s − 1.04·22-s + 0.701·23-s + 24-s + 1.33·25-s + 0.0694·26-s − 27-s + 4.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.12·5-s + 0.408·6-s + 1.67·7-s − 0.353·8-s + 0.333·9-s − 0.796·10-s + 0.314·11-s − 0.288·12-s − 0.0192·13-s − 1.18·14-s − 0.649·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.0278·19-s + 0.562·20-s − 0.966·21-s − 0.222·22-s + 0.146·23-s + 0.204·24-s + 0.267·25-s + 0.0136·26-s − 0.192·27-s + 0.836·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.890654883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890654883\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 + 0.0694T + 13T^{2} \) |
| 19 | \( 1 - 0.121T + 19T^{2} \) |
| 23 | \( 1 - 0.701T + 23T^{2} \) |
| 29 | \( 1 + 7.77T + 29T^{2} \) |
| 31 | \( 1 + 5.56T + 31T^{2} \) |
| 37 | \( 1 + 0.224T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 8.54T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 7.35T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 7.20T + 83T^{2} \) |
| 89 | \( 1 - 1.35T + 89T^{2} \) |
| 97 | \( 1 - 3.72T + 97T^{2} \) |
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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
−7.999648972745545309759613214243, −7.49586715941775326860270830670, −6.69701966040649733990149426643, −5.86716899978841000972165584134, −5.39449218090388928649032856897, −4.66903591104686674344092168660, −3.68775085280241256649958091001, −2.17263108204109544275346973771, −1.85006164052778256076019243859, −0.865074412963826817840833062495,
0.865074412963826817840833062495, 1.85006164052778256076019243859, 2.17263108204109544275346973771, 3.68775085280241256649958091001, 4.66903591104686674344092168660, 5.39449218090388928649032856897, 5.86716899978841000972165584134, 6.69701966040649733990149426643, 7.49586715941775326860270830670, 7.999648972745545309759613214243
