| L(s) = 1 | − 0.796·2-s − 3-s − 1.36·4-s − 2.25·5-s + 0.796·6-s + 2.67·8-s + 9-s + 1.79·10-s + 1.03·11-s + 1.36·12-s − 3.01·13-s + 2.25·15-s + 0.598·16-s + 6.43·17-s − 0.796·18-s − 5.89·19-s + 3.07·20-s − 0.820·22-s + 2.43·23-s − 2.67·24-s + 0.0823·25-s + 2.39·26-s − 27-s − 5.31·29-s − 1.79·30-s − 5.96·31-s − 5.83·32-s + ⋯ |
| L(s) = 1 | − 0.562·2-s − 0.577·3-s − 0.683·4-s − 1.00·5-s + 0.325·6-s + 0.947·8-s + 0.333·9-s + 0.567·10-s + 0.310·11-s + 0.394·12-s − 0.835·13-s + 0.582·15-s + 0.149·16-s + 1.56·17-s − 0.187·18-s − 1.35·19-s + 0.688·20-s − 0.174·22-s + 0.508·23-s − 0.547·24-s + 0.0164·25-s + 0.470·26-s − 0.192·27-s − 0.986·29-s − 0.327·30-s − 1.07·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.796T + 2T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 6.43T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 + 5.96T + 31T^{2} \) |
| 37 | \( 1 + 0.364T + 37T^{2} \) |
| 41 | \( 1 - 7.24T + 41T^{2} \) |
| 43 | \( 1 - 8.97T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 + 5.85T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 + 1.97T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
| 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
−7.57226343790988838615372409967, −7.26001287179972714277516762059, −6.13549011013892568662266903080, −5.41245636074129423194176754591, −4.66631466938108189998851881333, −4.03343548741116881115308851450, −3.40129050614081763590823569004, −2.01398681957452947909185950169, −0.874679802267478985987535638278, 0,
0.874679802267478985987535638278, 2.01398681957452947909185950169, 3.40129050614081763590823569004, 4.03343548741116881115308851450, 4.66631466938108189998851881333, 5.41245636074129423194176754591, 6.13549011013892568662266903080, 7.26001287179972714277516762059, 7.57226343790988838615372409967
