| L(s) = 1 | + 2.84·2-s + 0.120·4-s − 12.8·5-s + 26.0·7-s − 22.4·8-s − 36.6·10-s + 62.8·11-s + 22.3·13-s + 74.2·14-s − 64.9·16-s + 57.9·17-s + 71.3·19-s − 1.55·20-s + 179.·22-s + 49.5·23-s + 40.7·25-s + 63.8·26-s + 3.15·28-s + 29·29-s + 62.9·31-s − 5.47·32-s + 165.·34-s − 335.·35-s + 119.·37-s + 203.·38-s + 289.·40-s + 414.·41-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.0151·4-s − 1.15·5-s + 1.40·7-s − 0.992·8-s − 1.16·10-s + 1.72·11-s + 0.477·13-s + 1.41·14-s − 1.01·16-s + 0.827·17-s + 0.861·19-s − 0.0174·20-s + 1.73·22-s + 0.449·23-s + 0.325·25-s + 0.481·26-s + 0.0212·28-s + 0.185·29-s + 0.364·31-s − 0.0302·32-s + 0.833·34-s − 1.61·35-s + 0.529·37-s + 0.867·38-s + 1.14·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.787124986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.787124986\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 2.84T + 8T^{2} \) |
| 5 | \( 1 + 12.8T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 11 | \( 1 - 62.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.5T + 1.21e4T^{2} \) |
| 31 | \( 1 - 62.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 348.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 553.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 579.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 919.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 133.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 868.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 83.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 357.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 187.T + 9.12e5T^{2} \) |
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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
−11.70153254789827150565043378863, −11.20370209264214927987409748517, −9.493053502675189443879691564725, −8.493045756016364363078188562632, −7.64074164259622965847602683762, −6.33368629537823133547948235087, −5.05527740302969023748672150192, −4.20425487845205432086156800022, −3.38360548610339223363721250476, −1.16441012272866089556179663158,
1.16441012272866089556179663158, 3.38360548610339223363721250476, 4.20425487845205432086156800022, 5.05527740302969023748672150192, 6.33368629537823133547948235087, 7.64074164259622965847602683762, 8.493045756016364363078188562632, 9.493053502675189443879691564725, 11.20370209264214927987409748517, 11.70153254789827150565043378863
