| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s − 2.82·5-s + 2.41·6-s − 4.41·8-s + 9-s + 6.82·10-s − 4.82·11-s − 3.82·12-s + 4.82·13-s + 2.82·15-s + 2.99·16-s − 1.17·17-s − 2.41·18-s + 19-s − 10.8·20-s + 11.6·22-s − 4.82·23-s + 4.41·24-s + 3.00·25-s − 11.6·26-s − 27-s + 8.48·29-s − 6.82·30-s + 1.58·32-s + 4.82·33-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s − 1.26·5-s + 0.985·6-s − 1.56·8-s + 0.333·9-s + 2.15·10-s − 1.45·11-s − 1.10·12-s + 1.33·13-s + 0.730·15-s + 0.749·16-s − 0.284·17-s − 0.569·18-s + 0.229·19-s − 2.42·20-s + 2.48·22-s − 1.00·23-s + 0.901·24-s + 0.600·25-s − 2.28·26-s − 0.192·27-s + 1.57·29-s − 1.24·30-s + 0.280·32-s + 0.840·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.410147778034792451501143801485, −7.86737967787523143635352192861, −7.22882538182510469482466646511, −6.46765489201639464633022690386, −5.54558731410674832945673475967, −4.44324317692325089445146164047, −3.43797314050492982962043905525, −2.27296157815587323033990197395, −0.951456954324631369735668202201, 0,
0.951456954324631369735668202201, 2.27296157815587323033990197395, 3.43797314050492982962043905525, 4.44324317692325089445146164047, 5.54558731410674832945673475967, 6.46765489201639464633022690386, 7.22882538182510469482466646511, 7.86737967787523143635352192861, 8.410147778034792451501143801485
