| L(s) = 1 | − 0.618·2-s − 2.61·3-s − 1.61·4-s − 5-s + 1.61·6-s + 2.23·8-s + 3.85·9-s + 0.618·10-s − 1.61·11-s + 4.23·12-s + 13-s + 2.61·15-s + 1.85·16-s + 2.85·17-s − 2.38·18-s + 19-s + 1.61·20-s + 1.00·22-s + 3.47·23-s − 5.85·24-s − 4·25-s − 0.618·26-s − 2.23·27-s + 3.61·29-s − 1.61·30-s + 10.5·31-s − 5.61·32-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 1.51·3-s − 0.809·4-s − 0.447·5-s + 0.660·6-s + 0.790·8-s + 1.28·9-s + 0.195·10-s − 0.487·11-s + 1.22·12-s + 0.277·13-s + 0.675·15-s + 0.463·16-s + 0.692·17-s − 0.561·18-s + 0.229·19-s + 0.361·20-s + 0.213·22-s + 0.723·23-s − 1.19·24-s − 0.800·25-s − 0.121·26-s − 0.430·27-s + 0.671·29-s − 0.295·30-s + 1.89·31-s − 0.993·32-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 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 1.85T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 5.94T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 4.70T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 9.85T + 83T^{2} \) |
| 89 | \( 1 + 7.23T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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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
−10.01743829146338455472767099287, −8.764520872120321610601875903152, −8.017137261804027839550424637821, −7.09442558566743820221018968445, −6.10160260701595760390970412024, −5.15637072280566123317405722388, −4.63324355249392021220879636521, −3.38663266594417518382502288044, −1.21307867701677248734773070883, 0,
1.21307867701677248734773070883, 3.38663266594417518382502288044, 4.63324355249392021220879636521, 5.15637072280566123317405722388, 6.10160260701595760390970412024, 7.09442558566743820221018968445, 8.017137261804027839550424637821, 8.764520872120321610601875903152, 10.01743829146338455472767099287
