| L(s) = 1 | − 1.07·3-s − 11.6·5-s − 2.67·7-s − 25.8·9-s + 63.9·11-s − 50.0·13-s + 12.4·15-s + 72.4·17-s + 27.4·19-s + 2.85·21-s + 139.·23-s + 10.3·25-s + 56.5·27-s + 93.3·29-s − 188.·31-s − 68.4·33-s + 31.0·35-s + 118.·37-s + 53.5·39-s + 104.·41-s + 44.5·43-s + 300.·45-s − 488.·47-s − 335.·49-s − 77.5·51-s + 211.·53-s − 743.·55-s + ⋯ |
| L(s) = 1 | − 0.205·3-s − 1.04·5-s − 0.144·7-s − 0.957·9-s + 1.75·11-s − 1.06·13-s + 0.214·15-s + 1.03·17-s + 0.332·19-s + 0.0297·21-s + 1.26·23-s + 0.0826·25-s + 0.403·27-s + 0.598·29-s − 1.09·31-s − 0.361·33-s + 0.150·35-s + 0.528·37-s + 0.219·39-s + 0.398·41-s + 0.157·43-s + 0.996·45-s − 1.51·47-s − 0.979·49-s − 0.213·51-s + 0.548·53-s − 1.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.07T + 27T^{2} \) |
| 5 | \( 1 + 11.6T + 125T^{2} \) |
| 7 | \( 1 + 2.67T + 343T^{2} \) |
| 11 | \( 1 - 63.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 93.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 44.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 402.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 196.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 259.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 797.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 866.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 936.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.165836954440399061194443154856, −8.311067818522518026779240541010, −7.43253475357908082505143561744, −6.72078552462434158475186592952, −5.68943511494886771565955266150, −4.71532194653562024482271713198, −3.70573473816378447599706886080, −2.91414073020808539088861518470, −1.21674533576091590028503796600, 0,
1.21674533576091590028503796600, 2.91414073020808539088861518470, 3.70573473816378447599706886080, 4.71532194653562024482271713198, 5.68943511494886771565955266150, 6.72078552462434158475186592952, 7.43253475357908082505143561744, 8.311067818522518026779240541010, 9.165836954440399061194443154856
