| L(s) = 1 | − 2-s − 1.62·3-s + 4-s + 5-s + 1.62·6-s − 2.80·7-s − 8-s − 0.343·9-s − 10-s + 11-s − 1.62·12-s + 1.03·13-s + 2.80·14-s − 1.62·15-s + 16-s + 2.39·17-s + 0.343·18-s + 0.827·19-s + 20-s + 4.57·21-s − 22-s − 2.07·23-s + 1.62·24-s + 25-s − 1.03·26-s + 5.44·27-s − 2.80·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.941·3-s + 0.5·4-s + 0.447·5-s + 0.665·6-s − 1.06·7-s − 0.353·8-s − 0.114·9-s − 0.316·10-s + 0.301·11-s − 0.470·12-s + 0.288·13-s + 0.749·14-s − 0.420·15-s + 0.250·16-s + 0.582·17-s + 0.0808·18-s + 0.189·19-s + 0.223·20-s + 0.997·21-s − 0.213·22-s − 0.433·23-s + 0.332·24-s + 0.200·25-s − 0.203·26-s + 1.04·27-s − 0.530·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 + 1.62T + 3T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 - 0.827T + 19T^{2} \) |
| 23 | \( 1 + 2.07T + 23T^{2} \) |
| 29 | \( 1 + 2.69T + 29T^{2} \) |
| 31 | \( 1 + 0.0184T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 + 9.31T + 41T^{2} \) |
| 43 | \( 1 + 0.487T + 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 6.09T + 71T^{2} \) |
| 79 | \( 1 - 6.97T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 9.27T + 89T^{2} \) |
| 97 | \( 1 - 9.74T + 97T^{2} \) |
| 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
−7.39121445429250252635772913923, −6.61103407926199167788464446542, −6.25074198004757976328984575944, −5.61152265635941508714197419678, −4.94891337237873757333181572187, −3.71248494958622600704244019070, −3.08733204295801263150520236905, −2.03866324968351553276028319288, −0.975254778130705960411851470579, 0,
0.975254778130705960411851470579, 2.03866324968351553276028319288, 3.08733204295801263150520236905, 3.71248494958622600704244019070, 4.94891337237873757333181572187, 5.61152265635941508714197419678, 6.25074198004757976328984575944, 6.61103407926199167788464446542, 7.39121445429250252635772913923
