| L(s) = 1 | − 0.381·2-s − 1.61·3-s − 1.85·4-s + 5-s + 0.618·6-s + 7-s + 1.47·8-s − 0.381·9-s − 0.381·10-s + 3·12-s − 1.23·13-s − 0.381·14-s − 1.61·15-s + 3.14·16-s + 3.09·17-s + 0.145·18-s − 1.76·19-s − 1.85·20-s − 1.61·21-s + 5.09·23-s − 2.38·24-s − 4·25-s + 0.472·26-s + 5.47·27-s − 1.85·28-s − 4.61·29-s + 0.618·30-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.934·3-s − 0.927·4-s + 0.447·5-s + 0.252·6-s + 0.377·7-s + 0.520·8-s − 0.127·9-s − 0.120·10-s + 0.866·12-s − 0.342·13-s − 0.102·14-s − 0.417·15-s + 0.786·16-s + 0.749·17-s + 0.0343·18-s − 0.404·19-s − 0.414·20-s − 0.353·21-s + 1.06·23-s − 0.486·24-s − 0.800·25-s + 0.0925·26-s + 1.05·27-s − 0.350·28-s − 0.857·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 3.09T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 + 8.32T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.865078960704782089208818551945, −8.984160913290049994533083852919, −8.191099614969721863271154062532, −7.22059825385241821925502660797, −6.05112249494461483758841946808, −5.30398997807561911981799181329, −4.67318066006327461208361749261, −3.33236426741763899825414396496, −1.55579214088232074399455203926, 0,
1.55579214088232074399455203926, 3.33236426741763899825414396496, 4.67318066006327461208361749261, 5.30398997807561911981799181329, 6.05112249494461483758841946808, 7.22059825385241821925502660797, 8.191099614969721863271154062532, 8.984160913290049994533083852919, 9.865078960704782089208818551945
