| L(s) = 1 | + 1.44·2-s + 1.55·3-s + 0.0881·4-s + 3.60·5-s + 2.24·6-s − 7-s − 2.76·8-s − 0.582·9-s + 5.20·10-s − 4.49·11-s + 0.137·12-s + 4.13·13-s − 1.44·14-s + 5.60·15-s − 4.16·16-s + 2.02·17-s − 0.841·18-s − 8.29·19-s + 0.317·20-s − 1.55·21-s − 6.49·22-s − 1.06·23-s − 4.29·24-s + 7.98·25-s + 5.97·26-s − 5.57·27-s − 0.0881·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.897·3-s + 0.0440·4-s + 1.61·5-s + 0.917·6-s − 0.377·7-s − 0.976·8-s − 0.194·9-s + 1.64·10-s − 1.35·11-s + 0.0395·12-s + 1.14·13-s − 0.386·14-s + 1.44·15-s − 1.04·16-s + 0.490·17-s − 0.198·18-s − 1.90·19-s + 0.0710·20-s − 0.339·21-s − 1.38·22-s − 0.221·23-s − 0.876·24-s + 1.59·25-s + 1.17·26-s − 1.07·27-s − 0.0166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.681161139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.681161139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 - 3.60T + 5T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 43 | \( 1 + 0.515T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 + 3.50T + 53T^{2} \) |
| 59 | \( 1 + 0.933T + 59T^{2} \) |
| 61 | \( 1 + 6.59T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 + 0.219T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 0.841T + 97T^{2} \) |
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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
−12.32386462072835217944061741480, −10.69461387166473648424526864350, −9.954001284091739421027746833563, −8.867685696442434392443912921090, −8.280674905981217111329623124248, −6.33501450163292135971354037291, −5.85793649637084816263012440985, −4.65461081375038253673020373116, −3.16354792183047857882652915647, −2.33837052397192176419324146806,
2.33837052397192176419324146806, 3.16354792183047857882652915647, 4.65461081375038253673020373116, 5.85793649637084816263012440985, 6.33501450163292135971354037291, 8.280674905981217111329623124248, 8.867685696442434392443912921090, 9.954001284091739421027746833563, 10.69461387166473648424526864350, 12.32386462072835217944061741480
