| L(s) = 1 | + 0.414·3-s − 0.414·7-s − 2.82·9-s − 1.41·11-s + 3.82·13-s − 17-s − 19-s − 0.171·21-s + 3.24·23-s − 2.41·27-s − 1.82·29-s + 0.585·31-s − 0.585·33-s − 10.8·37-s + 1.58·39-s + 7.07·41-s + 6.24·43-s − 8·47-s − 6.82·49-s − 0.414·51-s − 3.82·53-s − 0.414·57-s + 11.5·59-s + 0.585·61-s + 1.17·63-s − 8.07·67-s + 1.34·69-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 0.156·7-s − 0.942·9-s − 0.426·11-s + 1.06·13-s − 0.242·17-s − 0.229·19-s − 0.0374·21-s + 0.676·23-s − 0.464·27-s − 0.339·29-s + 0.105·31-s − 0.101·33-s − 1.78·37-s + 0.253·39-s + 1.10·41-s + 0.951·43-s − 1.16·47-s − 0.975·49-s − 0.0580·51-s − 0.525·53-s − 0.0548·57-s + 1.50·59-s + 0.0750·61-s + 0.147·63-s − 0.986·67-s + 0.161·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 - 0.585T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.585T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 8.17T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.315221428175077604275083984708, −7.44833293480876185441432400712, −6.62042612143911151988337308163, −5.86981534874639817638277356519, −5.24012680313581551766316391863, −4.20936303510805103797572586019, −3.32736813535405626143985655363, −2.64206908699182052156205365259, −1.47408310659442908198348667766, 0,
1.47408310659442908198348667766, 2.64206908699182052156205365259, 3.32736813535405626143985655363, 4.20936303510805103797572586019, 5.24012680313581551766316391863, 5.86981534874639817638277356519, 6.62042612143911151988337308163, 7.44833293480876185441432400712, 8.315221428175077604275083984708
