| L(s) = 1 | + 1.23·5-s + 7-s − 3.01·11-s − 2.87·13-s + 2.65·17-s − 19-s − 4.87·23-s − 3.47·25-s − 0.359·29-s + 7.89·31-s + 1.23·35-s + 3.86·37-s − 5.28·41-s + 9.75·43-s − 0.222·47-s + 49-s − 3.64·53-s − 3.72·55-s + 2.02·59-s + 7.75·61-s − 3.55·65-s + 3.01·67-s − 7.30·71-s − 7.30·73-s − 3.01·77-s − 7.62·79-s + 1.19·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s + 0.377·7-s − 0.908·11-s − 0.797·13-s + 0.643·17-s − 0.229·19-s − 1.01·23-s − 0.694·25-s − 0.0667·29-s + 1.41·31-s + 0.208·35-s + 0.635·37-s − 0.824·41-s + 1.48·43-s − 0.0324·47-s + 0.142·49-s − 0.500·53-s − 0.502·55-s + 0.263·59-s + 0.992·61-s − 0.441·65-s + 0.368·67-s − 0.866·71-s − 0.855·73-s − 0.343·77-s − 0.857·79-s + 0.131·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 3.01T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 23 | \( 1 + 4.87T + 23T^{2} \) |
| 29 | \( 1 + 0.359T + 29T^{2} \) |
| 31 | \( 1 - 7.89T + 31T^{2} \) |
| 37 | \( 1 - 3.86T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 - 9.75T + 43T^{2} \) |
| 47 | \( 1 + 0.222T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 7.30T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 1.19T + 83T^{2} \) |
| 89 | \( 1 - 7.30T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.49892619447219127353315863416, −6.63218872087128993386602625490, −5.85370056408549849442377649028, −5.39209877737626709519082149691, −4.61748802426789535858817413849, −3.92324132623180766970903271121, −2.76964688819261920341604991698, −2.30773294879294092255452242229, −1.29820591223838438844353659831, 0,
1.29820591223838438844353659831, 2.30773294879294092255452242229, 2.76964688819261920341604991698, 3.92324132623180766970903271121, 4.61748802426789535858817413849, 5.39209877737626709519082149691, 5.85370056408549849442377649028, 6.63218872087128993386602625490, 7.49892619447219127353315863416
