| L(s) = 1 | − 0.404·2-s − 7.83·4-s + 0.184·5-s + 12.9·7-s + 6.40·8-s − 0.0746·10-s + 9.52·11-s − 51.2·13-s − 5.21·14-s + 60.0·16-s − 40.8·17-s + 21.4·19-s − 1.44·20-s − 3.85·22-s − 16.4·23-s − 124.·25-s + 20.7·26-s − 101.·28-s + 181.·29-s + 56.6·31-s − 75.5·32-s + 16.5·34-s + 2.37·35-s − 3.43·37-s − 8.68·38-s + 1.18·40-s + 367.·41-s + ⋯ |
| L(s) = 1 | − 0.143·2-s − 0.979·4-s + 0.0164·5-s + 0.696·7-s + 0.283·8-s − 0.00235·10-s + 0.261·11-s − 1.09·13-s − 0.0996·14-s + 0.939·16-s − 0.582·17-s + 0.259·19-s − 0.0161·20-s − 0.0373·22-s − 0.149·23-s − 0.999·25-s + 0.156·26-s − 0.682·28-s + 1.16·29-s + 0.328·31-s − 0.417·32-s + 0.0833·34-s + 0.0114·35-s − 0.0152·37-s − 0.0370·38-s + 0.00467·40-s + 1.39·41-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
| \(L(2)\) |
\(\approx\) |
\(1.299932219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299932219\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.404T + 8T^{2} \) |
| 5 | \( 1 - 0.184T + 125T^{2} \) |
| 7 | \( 1 - 12.9T + 343T^{2} \) |
| 11 | \( 1 - 9.52T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 3.43T + 5.06e4T^{2} \) |
| 41 | \( 1 - 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 3.14T + 7.95e4T^{2} \) |
| 47 | \( 1 + 423.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 648.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 328.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 285.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 206.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 117.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 623.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 737.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 244.T + 9.12e5T^{2} \) |
| 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
−8.720299374630055240294255344555, −8.013183570576077095094152389121, −7.40836904276685611063399039808, −6.34166722683409658646847778061, −5.36885875135838708184467293800, −4.65311461952116147643745293333, −4.07691316360470016208884922696, −2.83100205075908555246171067576, −1.69333111358836905627382435050, −0.54221634305489775568465419821,
0.54221634305489775568465419821, 1.69333111358836905627382435050, 2.83100205075908555246171067576, 4.07691316360470016208884922696, 4.65311461952116147643745293333, 5.36885875135838708184467293800, 6.34166722683409658646847778061, 7.40836904276685611063399039808, 8.013183570576077095094152389121, 8.720299374630055240294255344555
