| L(s) = 1 | − 5-s − 3.44·7-s − 1.44·11-s + 5.89·13-s + 4.89·17-s − 4·19-s + 5.44·23-s − 4·25-s + 0.101·29-s − 2.55·31-s + 3.44·35-s − 0.898·37-s − 11.8·41-s + 2.34·43-s + 6.34·47-s + 4.89·49-s + 8.89·53-s + 1.44·55-s − 14.3·59-s − 7.89·61-s − 5.89·65-s − 12.3·67-s − 7.79·71-s − 4.89·73-s + 5·77-s − 13.4·79-s + 0.550·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.30·7-s − 0.437·11-s + 1.63·13-s + 1.18·17-s − 0.917·19-s + 1.13·23-s − 0.800·25-s + 0.0187·29-s − 0.458·31-s + 0.583·35-s − 0.147·37-s − 1.85·41-s + 0.358·43-s + 0.926·47-s + 0.699·49-s + 1.22·53-s + 0.195·55-s − 1.86·59-s − 1.01·61-s − 0.731·65-s − 1.50·67-s − 0.925·71-s − 0.573·73-s + 0.569·77-s − 1.51·79-s + 0.0604·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 5.44T + 23T^{2} \) |
| 29 | \( 1 - 0.101T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 + 0.898T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 7.89T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 + 7.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 0.550T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−8.633423619948055414646113075618, −7.70714549939815560863065586906, −6.97762241430200115177365292856, −6.11350928907050060360986926898, −5.61976924248753063720234790252, −4.34722039498800434535699851600, −3.49843136992129748803582935263, −2.97192937483187741065428773673, −1.42073272012521395322399751058, 0,
1.42073272012521395322399751058, 2.97192937483187741065428773673, 3.49843136992129748803582935263, 4.34722039498800434535699851600, 5.61976924248753063720234790252, 6.11350928907050060360986926898, 6.97762241430200115177365292856, 7.70714549939815560863065586906, 8.633423619948055414646113075618
