| L(s) = 1 | − 2.41·2-s + 1.41·3-s + 3.82·4-s + 5-s − 3.41·6-s − 4.41·8-s − 0.999·9-s − 2.41·10-s − 0.414·11-s + 5.41·12-s − 2.24·13-s + 1.41·15-s + 2.99·16-s − 4·17-s + 2.41·18-s − 19-s + 3.82·20-s + 0.999·22-s − 5.58·23-s − 6.24·24-s − 4·25-s + 5.41·26-s − 5.65·27-s − 6.58·29-s − 3.41·30-s − 6.24·31-s + 1.58·32-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.816·3-s + 1.91·4-s + 0.447·5-s − 1.39·6-s − 1.56·8-s − 0.333·9-s − 0.763·10-s − 0.124·11-s + 1.56·12-s − 0.621·13-s + 0.365·15-s + 0.749·16-s − 0.970·17-s + 0.569·18-s − 0.229·19-s + 0.856·20-s + 0.213·22-s − 1.16·23-s − 1.27·24-s − 0.800·25-s + 1.06·26-s − 1.08·27-s − 1.22·29-s − 0.623·30-s − 1.12·31-s + 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−9.456141203870085480666378544387, −8.980005771305112606958486486527, −8.056091000350368246949434551053, −7.60219746470525571015287112157, −6.55462625958151421936933079323, −5.59090544571134260529815484106, −3.96375046843835255065044519639, −2.48619729114377752456631670609, −1.94149399450553034089959645972, 0,
1.94149399450553034089959645972, 2.48619729114377752456631670609, 3.96375046843835255065044519639, 5.59090544571134260529815484106, 6.55462625958151421936933079323, 7.60219746470525571015287112157, 8.056091000350368246949434551053, 8.980005771305112606958486486527, 9.456141203870085480666378544387
