| L(s) = 1 | − 3-s − 2·9-s + 2·11-s − 4·17-s − 2·19-s − 23-s + 5·27-s + 9·29-s + 4·31-s − 2·33-s − 4·37-s + 41-s − 9·43-s + 4·51-s + 10·53-s + 2·57-s − 10·59-s + 9·61-s − 5·67-s + 69-s + 14·71-s − 12·73-s + 14·79-s + 81-s − 11·83-s − 9·87-s − 15·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.603·11-s − 0.970·17-s − 0.458·19-s − 0.208·23-s + 0.962·27-s + 1.67·29-s + 0.718·31-s − 0.348·33-s − 0.657·37-s + 0.156·41-s − 1.37·43-s + 0.560·51-s + 1.37·53-s + 0.264·57-s − 1.30·59-s + 1.15·61-s − 0.610·67-s + 0.120·69-s + 1.66·71-s − 1.40·73-s + 1.57·79-s + 1/9·81-s − 1.20·83-s − 0.964·87-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.11940382916562769380244118564, −6.49064321914241615745315723343, −6.17161502867647856199519252884, −5.21839308674570266688020793945, −4.66776323393633705224746462061, −3.90395022502470222098843188524, −2.97111346358414364452819966931, −2.20137090646183567088631138610, −1.07748740319160806845883403538, 0,
1.07748740319160806845883403538, 2.20137090646183567088631138610, 2.97111346358414364452819966931, 3.90395022502470222098843188524, 4.66776323393633705224746462061, 5.21839308674570266688020793945, 6.17161502867647856199519252884, 6.49064321914241615745315723343, 7.11940382916562769380244118564
