| L(s) = 1 | + 2.14·2-s + 3-s + 2.60·4-s − 2.36·5-s + 2.14·6-s + 1.30·8-s + 9-s − 5.07·10-s − 2.70·11-s + 2.60·12-s − 3.97·13-s − 2.36·15-s − 2.41·16-s + 6.48·17-s + 2.14·18-s + 7.54·19-s − 6.16·20-s − 5.79·22-s − 1.03·23-s + 1.30·24-s + 0.587·25-s − 8.52·26-s + 27-s − 8.83·29-s − 5.07·30-s − 2.51·31-s − 7.79·32-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 0.577·3-s + 1.30·4-s − 1.05·5-s + 0.876·6-s + 0.460·8-s + 0.333·9-s − 1.60·10-s − 0.814·11-s + 0.752·12-s − 1.10·13-s − 0.610·15-s − 0.604·16-s + 1.57·17-s + 0.505·18-s + 1.73·19-s − 1.37·20-s − 1.23·22-s − 0.216·23-s + 0.265·24-s + 0.117·25-s − 1.67·26-s + 0.192·27-s − 1.64·29-s − 0.926·30-s − 0.452·31-s − 1.37·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 - 7.54T + 19T^{2} \) |
| 23 | \( 1 + 1.03T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 43 | \( 1 + 0.547T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 + 8.41T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 4.26T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 + 6.72T + 71T^{2} \) |
| 73 | \( 1 - 7.72T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 7.45T + 97T^{2} \) |
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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
−7.56165395506026952916030406400, −7.25350648529688470630568925400, −6.03829619436180578271778370200, −5.24127225757662518980758508574, −4.92361690343460291417291667601, −3.89596672632467287991911004029, −3.33135417791800884988832775597, −2.89416711648602318226886373255, −1.72333876467311491672337357711, 0,
1.72333876467311491672337357711, 2.89416711648602318226886373255, 3.33135417791800884988832775597, 3.89596672632467287991911004029, 4.92361690343460291417291667601, 5.24127225757662518980758508574, 6.03829619436180578271778370200, 7.25350648529688470630568925400, 7.56165395506026952916030406400
