| L(s) = 1 | − 1.61·2-s − 1.61·3-s + 0.618·4-s − 0.381·5-s + 2.61·6-s + 7-s + 2.23·8-s − 0.381·9-s + 0.618·10-s + 2.23·11-s − 1.00·12-s − 0.381·13-s − 1.61·14-s + 0.618·15-s − 4.85·16-s + 3·17-s + 0.618·18-s − 1.76·19-s − 0.236·20-s − 1.61·21-s − 3.61·22-s − 7.47·23-s − 3.61·24-s − 4.85·25-s + 0.618·26-s + 5.47·27-s + 0.618·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.934·3-s + 0.309·4-s − 0.170·5-s + 1.06·6-s + 0.377·7-s + 0.790·8-s − 0.127·9-s + 0.195·10-s + 0.674·11-s − 0.288·12-s − 0.105·13-s − 0.432·14-s + 0.159·15-s − 1.21·16-s + 0.727·17-s + 0.145·18-s − 0.404·19-s − 0.0527·20-s − 0.353·21-s − 0.771·22-s − 1.55·23-s − 0.738·24-s − 0.970·25-s + 0.121·26-s + 1.05·27-s + 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.381T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 6.61T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 1.47T + 73T^{2} \) |
| 79 | \( 1 + 8.38T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 - 5.94T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−10.72657444532132426751397079920, −10.13170578664960710445196465044, −9.078276867156160750853000508807, −8.239852845016159817608601191966, −7.35239868217304736200478019461, −6.20145498572491495100158933378, −5.15857337828492853312134083077, −3.91086095617363628627255719335, −1.69526603202017564021313689411, 0,
1.69526603202017564021313689411, 3.91086095617363628627255719335, 5.15857337828492853312134083077, 6.20145498572491495100158933378, 7.35239868217304736200478019461, 8.239852845016159817608601191966, 9.078276867156160750853000508807, 10.13170578664960710445196465044, 10.72657444532132426751397079920
