| L(s) = 1 | − 2-s + 3-s + 4-s − 1.73·5-s − 6-s − 3.07·7-s − 8-s + 9-s + 1.73·10-s − 1.39·11-s + 12-s − 13-s + 3.07·14-s − 1.73·15-s + 16-s − 2.86·17-s − 18-s + 7.95·19-s − 1.73·20-s − 3.07·21-s + 1.39·22-s + 7.53·23-s − 24-s − 1.98·25-s + 26-s + 27-s − 3.07·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.776·5-s − 0.408·6-s − 1.16·7-s − 0.353·8-s + 0.333·9-s + 0.548·10-s − 0.419·11-s + 0.288·12-s − 0.277·13-s + 0.821·14-s − 0.448·15-s + 0.250·16-s − 0.694·17-s − 0.235·18-s + 1.82·19-s − 0.388·20-s − 0.671·21-s + 0.296·22-s + 1.57·23-s − 0.204·24-s − 0.397·25-s + 0.196·26-s + 0.192·27-s − 0.581·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 - 7.95T + 19T^{2} \) |
| 23 | \( 1 - 7.53T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 + 2.18T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 41 | \( 1 + 9.35T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 - 2.12T + 53T^{2} \) |
| 59 | \( 1 + 8.76T + 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 - 1.69T + 67T^{2} \) |
| 71 | \( 1 - 7.75T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.35T + 83T^{2} \) |
| 89 | \( 1 - 2.39T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.52348734309488637856309568949, −7.01742323992361985751323108851, −6.43940873596427807547213875674, −5.38344827435516851169284002013, −4.64794872121918258851786058292, −3.43054659827374999660483343589, −3.23205005510216568665041350903, −2.32696280735106909493050351957, −1.06282849101535722597941284503, 0,
1.06282849101535722597941284503, 2.32696280735106909493050351957, 3.23205005510216568665041350903, 3.43054659827374999660483343589, 4.64794872121918258851786058292, 5.38344827435516851169284002013, 6.43940873596427807547213875674, 7.01742323992361985751323108851, 7.52348734309488637856309568949
