L(s) = 1 | − 2.41·2-s − 2.17·4-s − 19.8·5-s + 24.5·8-s + 48.0·10-s − 23.9·11-s + 87.3·13-s − 41.9·16-s + 5.63·17-s + 64.8·19-s + 43.2·20-s + 57.7·22-s + 25.5·23-s + 270.·25-s − 210.·26-s − 60.3·29-s − 122.·31-s − 95.2·32-s − 13.6·34-s − 56.1·37-s − 156.·38-s − 488.·40-s + 299.·41-s − 501.·43-s + 51.9·44-s − 61.7·46-s − 305.·47-s + ⋯ |
L(s) = 1 | − 0.853·2-s − 0.271·4-s − 1.77·5-s + 1.08·8-s + 1.51·10-s − 0.656·11-s + 1.86·13-s − 0.654·16-s + 0.0804·17-s + 0.783·19-s + 0.483·20-s + 0.560·22-s + 0.232·23-s + 2.16·25-s − 1.59·26-s − 0.386·29-s − 0.710·31-s − 0.526·32-s − 0.0686·34-s − 0.249·37-s − 0.668·38-s − 1.93·40-s + 1.14·41-s − 1.77·43-s + 0.178·44-s − 0.198·46-s − 0.948·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.41T + 8T^{2} \) |
| 5 | \( 1 + 19.8T + 125T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 87.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.63T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 501.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 3.75T + 2.26e5T^{2} \) |
| 67 | \( 1 + 813.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 138.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 621.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 285.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 603.T + 9.12e5T^{2} \) |
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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
−10.33419811440724983581609669394, −9.088274030991434728730107760875, −8.357014619601616012426625719252, −7.82400630255851885196546771068, −6.93810883557655921831242610460, −5.33814560814580709424506664377, −4.14569041347904889351715367624, −3.37722073266509372629783289847, −1.16261129241548893594957761267, 0,
1.16261129241548893594957761267, 3.37722073266509372629783289847, 4.14569041347904889351715367624, 5.33814560814580709424506664377, 6.93810883557655921831242610460, 7.82400630255851885196546771068, 8.357014619601616012426625719252, 9.088274030991434728730107760875, 10.33419811440724983581609669394