| L(s) = 1 | + 3-s + 9-s − 2·11-s − 4·13-s + 6·17-s + 2·19-s + 27-s − 2·29-s + 4·31-s − 2·33-s + 2·37-s − 4·39-s − 2·41-s + 43-s − 8·47-s − 7·49-s + 6·51-s − 2·53-s + 2·57-s + 2·59-s − 2·61-s − 8·67-s − 4·71-s + 10·73-s + 12·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.328·37-s − 0.640·39-s − 0.312·41-s + 0.152·43-s − 1.16·47-s − 49-s + 0.840·51-s − 0.274·53-s + 0.264·57-s + 0.260·59-s − 0.256·61-s − 0.977·67-s − 0.474·71-s + 1.17·73-s + 1.35·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.78811336927425, −14.27231637824701, −13.77280913921850, −13.23912536686514, −12.72571645920978, −12.18249864810573, −11.83399242130822, −11.14812114886328, −10.42510628759104, −10.07683549618429, −9.486926276906813, −9.244350718573665, −8.224868886633569, −7.932887824639291, −7.574228096276546, −6.865636154886933, −6.299803320113645, −5.448566408779979, −5.092595913395734, −4.484685585496855, −3.648144791444585, −3.092796617622982, −2.597892432348523, −1.812127254812336, −1.032918316041234, 0,
1.032918316041234, 1.812127254812336, 2.597892432348523, 3.092796617622982, 3.648144791444585, 4.484685585496855, 5.092595913395734, 5.448566408779979, 6.299803320113645, 6.865636154886933, 7.574228096276546, 7.932887824639291, 8.224868886633569, 9.244350718573665, 9.486926276906813, 10.07683549618429, 10.42510628759104, 11.14812114886328, 11.83399242130822, 12.18249864810573, 12.72571645920978, 13.23912536686514, 13.77280913921850, 14.27231637824701, 14.78811336927425
