| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s − 2.82·13-s − 15-s − 1.17·17-s + 5.41·19-s − 7.41·23-s + 25-s + 27-s + 3.65·29-s + 4.24·31-s − 2·33-s − 10.4·37-s − 2.82·39-s − 2·41-s + 1.65·43-s − 45-s − 10.4·47-s − 1.17·51-s − 4.58·53-s + 2·55-s + 5.41·57-s + 2.82·59-s − 9.89·61-s + 2.82·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.333·9-s − 0.603·11-s − 0.784·13-s − 0.258·15-s − 0.284·17-s + 1.24·19-s − 1.54·23-s + 0.200·25-s + 0.192·27-s + 0.679·29-s + 0.762·31-s − 0.348·33-s − 1.72·37-s − 0.452·39-s − 0.312·41-s + 0.252·43-s − 0.149·45-s − 1.52·47-s − 0.164·51-s − 0.629·53-s + 0.269·55-s + 0.717·57-s + 0.368·59-s − 1.26·61-s + 0.350·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 9.89T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.285524153546110888381137033724, −7.69869618968846624167442836280, −7.08892776025129797945443556212, −6.14019359025795660043162292327, −5.11646495457411816432896149745, −4.46920918228590367970698165872, −3.42783310173165847766412567750, −2.72784758884602205655690429024, −1.62797282457114292265469852673, 0,
1.62797282457114292265469852673, 2.72784758884602205655690429024, 3.42783310173165847766412567750, 4.46920918228590367970698165872, 5.11646495457411816432896149745, 6.14019359025795660043162292327, 7.08892776025129797945443556212, 7.69869618968846624167442836280, 8.285524153546110888381137033724
