| L(s) = 1 | + 3-s − 5-s − 1.82·7-s + 9-s + 2.58·11-s + 3.41·13-s − 15-s − 6.07·17-s + 6.24·19-s − 1.82·21-s + 23-s + 25-s + 27-s − 3.58·29-s + 4.17·31-s + 2.58·33-s + 1.82·35-s + 3·37-s + 3.41·39-s − 10.4·41-s − 6·43-s − 45-s + 8.58·47-s − 3.65·49-s − 6.07·51-s + 7.24·53-s − 2.58·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.691·7-s + 0.333·9-s + 0.779·11-s + 0.946·13-s − 0.258·15-s − 1.47·17-s + 1.43·19-s − 0.398·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.665·29-s + 0.749·31-s + 0.450·33-s + 0.309·35-s + 0.493·37-s + 0.546·39-s − 1.62·41-s − 0.914·43-s − 0.149·45-s + 1.25·47-s − 0.522·49-s − 0.850·51-s + 0.994·53-s − 0.348·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.203217037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203217037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 29 | \( 1 + 3.58T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 7.24T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−8.364869358971764432486544232724, −7.33564887958641428986346291676, −6.82265157402010568278774762325, −6.19398484201687217814882751083, −5.21620180852674665440509122172, −4.25640189290561918587938977529, −3.63035693658830653858738988437, −3.00821290859783217813905802207, −1.90587536437901508729160348106, −0.78400020226473795810350419120,
0.78400020226473795810350419120, 1.90587536437901508729160348106, 3.00821290859783217813905802207, 3.63035693658830653858738988437, 4.25640189290561918587938977529, 5.21620180852674665440509122172, 6.19398484201687217814882751083, 6.82265157402010568278774762325, 7.33564887958641428986346291676, 8.364869358971764432486544232724
