| L(s) = 1 | + 3-s − 1.23·5-s + 5.23·7-s + 9-s − 4.47·11-s + 13-s − 1.23·15-s + 4.47·17-s + 1.23·19-s + 5.23·21-s − 2.47·23-s − 3.47·25-s + 27-s + 8.47·29-s + 9.23·31-s − 4.47·33-s − 6.47·35-s + 4.47·37-s + 39-s − 9.23·41-s − 6.47·43-s − 1.23·45-s − 0.472·47-s + 20.4·49-s + 4.47·51-s − 0.472·53-s + 5.52·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.552·5-s + 1.97·7-s + 0.333·9-s − 1.34·11-s + 0.277·13-s − 0.319·15-s + 1.08·17-s + 0.283·19-s + 1.14·21-s − 0.515·23-s − 0.694·25-s + 0.192·27-s + 1.57·29-s + 1.65·31-s − 0.778·33-s − 1.09·35-s + 0.735·37-s + 0.160·39-s − 1.44·41-s − 0.986·43-s − 0.184·45-s − 0.0688·47-s + 2.91·49-s + 0.626·51-s − 0.0648·53-s + 0.745·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.557758759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.557758759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 5.23T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 9.23T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 0.472T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 6.94T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.404731768313652528679406572005, −8.090910383469264661923953285597, −7.897424478824379129418968213116, −6.82697863347255329343877043910, −5.54462597498153770754026702235, −4.91689003641409807824676541366, −4.21314400324694998902769330758, −3.10816359616780796317139973345, −2.15863573543653166004849330247, −1.05988439166724717835579986904,
1.05988439166724717835579986904, 2.15863573543653166004849330247, 3.10816359616780796317139973345, 4.21314400324694998902769330758, 4.91689003641409807824676541366, 5.54462597498153770754026702235, 6.82697863347255329343877043910, 7.897424478824379129418968213116, 8.090910383469264661923953285597, 8.404731768313652528679406572005
