| L(s) = 1 | − 4.78·2-s + 14.8·4-s + 19.8·5-s + 9.72·7-s − 32.9·8-s − 94.9·10-s + 8.54·11-s + 70.2·13-s − 46.5·14-s + 38.7·16-s + 97.0·17-s + 72.5·19-s + 295.·20-s − 40.8·22-s − 147.·23-s + 268.·25-s − 336.·26-s + 144.·28-s − 33.0·29-s + 113.·31-s + 78.7·32-s − 464.·34-s + 192.·35-s + 185.·37-s − 347.·38-s − 654.·40-s − 347.·41-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.86·4-s + 1.77·5-s + 0.525·7-s − 1.45·8-s − 3.00·10-s + 0.234·11-s + 1.49·13-s − 0.888·14-s + 0.604·16-s + 1.38·17-s + 0.876·19-s + 3.30·20-s − 0.396·22-s − 1.33·23-s + 2.15·25-s − 2.53·26-s + 0.977·28-s − 0.211·29-s + 0.655·31-s + 0.435·32-s − 2.34·34-s + 0.932·35-s + 0.822·37-s − 1.48·38-s − 2.58·40-s − 1.32·41-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
| \(L(2)\) |
\(\approx\) |
\(2.043133203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043133203\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.78T + 8T^{2} \) |
| 5 | \( 1 - 19.8T + 125T^{2} \) |
| 7 | \( 1 - 9.72T + 343T^{2} \) |
| 11 | \( 1 - 8.54T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 72.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 185.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 550.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 103.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 530.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 549.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 934.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 786.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 741.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 611.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.854812265836675576285381740486, −8.125084289193066261562784583152, −7.48763808310492647973616606527, −6.22964677163605686863556071629, −6.09219465313210106517887136037, −5.01432579670368406222054531327, −3.44484905245787659698468664339, −2.27741459528022370066455267211, −1.46169831800755629179676917087, −0.969662311429847364838524028651,
0.969662311429847364838524028651, 1.46169831800755629179676917087, 2.27741459528022370066455267211, 3.44484905245787659698468664339, 5.01432579670368406222054531327, 6.09219465313210106517887136037, 6.22964677163605686863556071629, 7.48763808310492647973616606527, 8.125084289193066261562784583152, 8.854812265836675576285381740486
