| L(s) = 1 | − 0.719·2-s − 1.48·4-s + 1.41·5-s − 0.0123·7-s + 2.50·8-s − 1.02·10-s + 2.65·11-s + 4.86·13-s + 0.00885·14-s + 1.16·16-s − 3.81·17-s + 5.63·19-s − 2.10·20-s − 1.90·22-s − 1.61·23-s − 2.98·25-s − 3.50·26-s + 0.0182·28-s − 2.29·29-s + 10.1·31-s − 5.84·32-s + 2.74·34-s − 0.0174·35-s − 5.34·37-s − 4.05·38-s + 3.55·40-s + 8.88·41-s + ⋯ |
| L(s) = 1 | − 0.508·2-s − 0.741·4-s + 0.634·5-s − 0.00464·7-s + 0.885·8-s − 0.323·10-s + 0.799·11-s + 1.35·13-s + 0.00236·14-s + 0.290·16-s − 0.924·17-s + 1.29·19-s − 0.470·20-s − 0.406·22-s − 0.335·23-s − 0.596·25-s − 0.687·26-s + 0.00344·28-s − 0.426·29-s + 1.82·31-s − 1.03·32-s + 0.470·34-s − 0.00295·35-s − 0.879·37-s − 0.657·38-s + 0.562·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.424232695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424232695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.719T + 2T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 0.0123T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 - 4.86T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 0.287T + 43T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 + 2.50T + 53T^{2} \) |
| 59 | \( 1 + 8.64T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 6.06T + 67T^{2} \) |
| 71 | \( 1 - 1.45T + 71T^{2} \) |
| 73 | \( 1 - 0.627T + 73T^{2} \) |
| 79 | \( 1 - 7.83T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 1.72T + 89T^{2} \) |
| 97 | \( 1 + 4.56T + 97T^{2} \) |
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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
−9.198977467852435044941606926189, −8.420481800318198016153570712564, −7.78952084857946517467844757553, −6.65090253898230501090013224072, −6.02829185984921354163459202014, −5.08121526584122643754635603710, −4.18089561898085281968995805149, −3.36657150504258448105358402328, −1.86321240347838209672644752716, −0.912610241298873622151152542190,
0.912610241298873622151152542190, 1.86321240347838209672644752716, 3.36657150504258448105358402328, 4.18089561898085281968995805149, 5.08121526584122643754635603710, 6.02829185984921354163459202014, 6.65090253898230501090013224072, 7.78952084857946517467844757553, 8.420481800318198016153570712564, 9.198977467852435044941606926189
