| L(s) = 1 | + 2.48·2-s − 1.83·4-s + 6.58·5-s + 7.57·7-s − 24.4·8-s + 16.3·10-s − 55.6·11-s + 9.79·13-s + 18.8·14-s − 45.9·16-s − 23.2·17-s + 55.5·19-s − 12.0·20-s − 138.·22-s + 4.24·23-s − 81.5·25-s + 24.3·26-s − 13.8·28-s + 164.·29-s + 26.4·31-s + 81.0·32-s − 57.7·34-s + 49.9·35-s + 138.·37-s + 138.·38-s − 160.·40-s − 377.·41-s + ⋯ |
| L(s) = 1 | + 0.878·2-s − 0.228·4-s + 0.589·5-s + 0.409·7-s − 1.07·8-s + 0.517·10-s − 1.52·11-s + 0.209·13-s + 0.359·14-s − 0.718·16-s − 0.331·17-s + 0.671·19-s − 0.134·20-s − 1.33·22-s + 0.0384·23-s − 0.652·25-s + 0.183·26-s − 0.0936·28-s + 1.05·29-s + 0.153·31-s + 0.447·32-s − 0.291·34-s + 0.241·35-s + 0.614·37-s + 0.589·38-s − 0.635·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.740199751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.740199751\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 2.48T + 8T^{2} \) |
| 5 | \( 1 - 6.58T + 125T^{2} \) |
| 7 | \( 1 - 7.57T + 343T^{2} \) |
| 11 | \( 1 + 55.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.79T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 4.24T + 1.21e4T^{2} \) |
| 29 | \( 1 - 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 26.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 377.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 24.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 234.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 12.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 741.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 13.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 818.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 487.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 889.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 576.T + 9.12e5T^{2} \) |
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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
−8.536920787275722121433939436576, −8.120322055047023888923402846856, −7.00950263228527667171203974648, −6.09775577883294848063526342664, −5.30307416090505404520154979537, −4.94725130497851299547687026872, −3.90632498311335311397235655420, −2.92774555519342824485611475003, −2.12567226519252499552162433703, −0.64010331618561346990256221961,
0.64010331618561346990256221961, 2.12567226519252499552162433703, 2.92774555519342824485611475003, 3.90632498311335311397235655420, 4.94725130497851299547687026872, 5.30307416090505404520154979537, 6.09775577883294848063526342664, 7.00950263228527667171203974648, 8.120322055047023888923402846856, 8.536920787275722121433939436576
