| L(s) = 1 | − 1.66·3-s − 1.44·5-s + 4.59·7-s − 0.239·9-s − 2.58·13-s + 2.40·15-s + 1.76·17-s − 19-s − 7.63·21-s + 3.19·23-s − 2.90·25-s + 5.38·27-s + 9.13·29-s − 8.95·31-s − 6.65·35-s − 11.2·37-s + 4.30·39-s − 3.21·41-s + 9.73·43-s + 0.346·45-s + 4.06·47-s + 14.1·49-s − 2.93·51-s − 7.05·53-s + 1.66·57-s − 3.60·59-s − 3.92·61-s + ⋯ |
| L(s) = 1 | − 0.959·3-s − 0.647·5-s + 1.73·7-s − 0.0797·9-s − 0.718·13-s + 0.621·15-s + 0.428·17-s − 0.229·19-s − 1.66·21-s + 0.666·23-s − 0.580·25-s + 1.03·27-s + 1.69·29-s − 1.60·31-s − 1.12·35-s − 1.85·37-s + 0.688·39-s − 0.501·41-s + 1.48·43-s + 0.0516·45-s + 0.593·47-s + 2.01·49-s − 0.411·51-s − 0.968·53-s + 0.220·57-s − 0.469·59-s − 0.502·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.66T + 3T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 29 | \( 1 - 9.13T + 29T^{2} \) |
| 31 | \( 1 + 8.95T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 - 9.73T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 + 3.92T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 3.59T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 1.56T + 97T^{2} \) |
| 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
−7.42903755297767855606821245246, −6.78040984390765606035095802619, −5.81151389731054732822735497465, −5.21626104718072576264019469167, −4.78206180981360300852576782399, −4.09540472410643348059595334735, −3.06396046310134759518014181074, −2.02075905000205630434553115490, −1.11166742571010440030493122417, 0,
1.11166742571010440030493122417, 2.02075905000205630434553115490, 3.06396046310134759518014181074, 4.09540472410643348059595334735, 4.78206180981360300852576782399, 5.21626104718072576264019469167, 5.81151389731054732822735497465, 6.78040984390765606035095802619, 7.42903755297767855606821245246
