| L(s) = 1 | + 1.41·3-s + 7-s − 0.999·9-s − 1.41·11-s − 1.41·19-s + 1.41·21-s − 4·23-s − 5·25-s − 5.65·27-s − 2.82·29-s − 4·31-s − 2.00·33-s − 2.82·37-s − 6·41-s − 4.24·43-s + 4·47-s + 49-s + 8.48·53-s − 2.00·57-s − 7.07·59-s + 5.65·61-s − 0.999·63-s + 1.41·67-s − 5.65·69-s + 8·71-s − 8·73-s − 7.07·75-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.377·7-s − 0.333·9-s − 0.426·11-s − 0.324·19-s + 0.308·21-s − 0.834·23-s − 25-s − 1.08·27-s − 0.525·29-s − 0.718·31-s − 0.348·33-s − 0.464·37-s − 0.937·41-s − 0.646·43-s + 0.583·47-s + 0.142·49-s + 1.16·53-s − 0.264·57-s − 0.920·59-s + 0.724·61-s − 0.125·63-s + 0.172·67-s − 0.681·69-s + 0.949·71-s − 0.936·73-s − 0.816·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 - 5.65T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.181942315457740223488288597996, −7.68904160207745781667481017932, −6.83395684037401519514740544051, −5.81137101406054218551036741457, −5.25305692870851908490684991727, −4.11877423261436345517750204851, −3.47914613175448171688505276291, −2.45058408135325349069273578301, −1.76054255250576206950985847152, 0,
1.76054255250576206950985847152, 2.45058408135325349069273578301, 3.47914613175448171688505276291, 4.11877423261436345517750204851, 5.25305692870851908490684991727, 5.81137101406054218551036741457, 6.83395684037401519514740544051, 7.68904160207745781667481017932, 8.181942315457740223488288597996
