L(s) = 1 | + 1.61·2-s + 0.381·3-s + 0.618·4-s + 0.618·6-s + 3·7-s − 2.23·8-s − 2.85·9-s − 3·11-s + 0.236·12-s + 4.85·13-s + 4.85·14-s − 4.85·16-s − 4.23·17-s − 4.61·18-s − 3.61·19-s + 1.14·21-s − 4.85·22-s + 1.23·23-s − 0.854·24-s + 7.85·26-s − 2.23·27-s + 1.85·28-s + 6.70·29-s + 5.09·31-s − 3.38·32-s − 1.14·33-s − 6.85·34-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.220·3-s + 0.309·4-s + 0.252·6-s + 1.13·7-s − 0.790·8-s − 0.951·9-s − 0.904·11-s + 0.0681·12-s + 1.34·13-s + 1.29·14-s − 1.21·16-s − 1.02·17-s − 1.08·18-s − 0.830·19-s + 0.250·21-s − 1.03·22-s + 0.257·23-s − 0.174·24-s + 1.54·26-s − 0.430·27-s + 0.350·28-s + 1.24·29-s + 0.914·31-s − 0.597·32-s − 0.199·33-s − 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.758926132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.758926132\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 6.09T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 0.472T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 7.85T + 97T^{2} \) |
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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
−13.66680448261143888113086115554, −12.59501013713967241204182460738, −11.43569191509064155717905656548, −10.74387534311400822453201533066, −8.808454161780028293555871773269, −8.239245164332923578427686594146, −6.38360348856968126184766725583, −5.28679580103268699123997621304, −4.21250092186211264242517437941, −2.64641955936591278226156008949,
2.64641955936591278226156008949, 4.21250092186211264242517437941, 5.28679580103268699123997621304, 6.38360348856968126184766725583, 8.239245164332923578427686594146, 8.808454161780028293555871773269, 10.74387534311400822453201533066, 11.43569191509064155717905656548, 12.59501013713967241204182460738, 13.66680448261143888113086115554