| L(s) = 1 | − 2-s + 2.83·3-s + 4-s + 5-s − 2.83·6-s − 1.19·7-s − 8-s + 5.02·9-s − 10-s − 11-s + 2.83·12-s + 2.20·13-s + 1.19·14-s + 2.83·15-s + 16-s + 0.322·17-s − 5.02·18-s − 7.74·19-s + 20-s − 3.39·21-s + 22-s − 4.63·23-s − 2.83·24-s + 25-s − 2.20·26-s + 5.73·27-s − 1.19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.63·3-s + 0.5·4-s + 0.447·5-s − 1.15·6-s − 0.452·7-s − 0.353·8-s + 1.67·9-s − 0.316·10-s − 0.301·11-s + 0.817·12-s + 0.611·13-s + 0.320·14-s + 0.731·15-s + 0.250·16-s + 0.0781·17-s − 1.18·18-s − 1.77·19-s + 0.223·20-s − 0.740·21-s + 0.213·22-s − 0.966·23-s − 0.578·24-s + 0.200·25-s − 0.432·26-s + 1.10·27-s − 0.226·28-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 | \( 1 - 2.83T + 3T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 - 0.322T + 17T^{2} \) |
| 19 | \( 1 + 7.74T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 - 0.505T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.46T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 5.41T + 53T^{2} \) |
| 59 | \( 1 + 4.63T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 + 8.89T + 71T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−7.65277414351865453554184427063, −7.09722669959006604811082748521, −6.23688654648402843999348237156, −5.64478307330799474946134622780, −4.31534227759845576343440625708, −3.68676184134116822520205572118, −2.95763485025048147613454798466, −2.07268994327572177848109156036, −1.69385371585986478085208140449, 0,
1.69385371585986478085208140449, 2.07268994327572177848109156036, 2.95763485025048147613454798466, 3.68676184134116822520205572118, 4.31534227759845576343440625708, 5.64478307330799474946134622780, 6.23688654648402843999348237156, 7.09722669959006604811082748521, 7.65277414351865453554184427063
