L(s) = 1 | + 2-s − 2.04·3-s + 4-s + 2.05·5-s − 2.04·6-s + 7-s + 8-s + 1.17·9-s + 2.05·10-s − 4.36·11-s − 2.04·12-s + 3.26·13-s + 14-s − 4.19·15-s + 16-s − 4.81·17-s + 1.17·18-s + 2.05·20-s − 2.04·21-s − 4.36·22-s − 0.277·23-s − 2.04·24-s − 0.789·25-s + 3.26·26-s + 3.73·27-s + 28-s − 6.08·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.17·3-s + 0.5·4-s + 0.917·5-s − 0.833·6-s + 0.377·7-s + 0.353·8-s + 0.390·9-s + 0.648·10-s − 1.31·11-s − 0.589·12-s + 0.906·13-s + 0.267·14-s − 1.08·15-s + 0.250·16-s − 1.16·17-s + 0.276·18-s + 0.458·20-s − 0.445·21-s − 0.930·22-s − 0.0578·23-s − 0.416·24-s − 0.157·25-s + 0.640·26-s + 0.718·27-s + 0.188·28-s − 1.13·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.231391241\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.231391241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 23 | \( 1 + 0.277T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 - 0.511T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 8.55T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 - 3.59T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 0.0848T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 9.44T + 79T^{2} \) |
| 83 | \( 1 - 6.18T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.131000651749614065110927152913, −7.28491579028955463480718034834, −6.43320658474589026181725433577, −5.87164483407313516051287777343, −5.49204968311330251047546075027, −4.74513139702237480548788981980, −4.01488802791724031826996831158, −2.70267408061728349552561220781, −2.05538812110474686837640960187, −0.76757167987104986092357914263,
0.76757167987104986092357914263, 2.05538812110474686837640960187, 2.70267408061728349552561220781, 4.01488802791724031826996831158, 4.74513139702237480548788981980, 5.49204968311330251047546075027, 5.87164483407313516051287777343, 6.43320658474589026181725433577, 7.28491579028955463480718034834, 8.131000651749614065110927152913