| L(s) = 1 | − 4.37·2-s + 11.1·4-s − 12.1·7-s − 13.6·8-s + 10.0·11-s + 48.5·13-s + 52.9·14-s − 29.3·16-s − 75.3·17-s − 116.·19-s − 43.8·22-s + 38.0·23-s − 212.·26-s − 134.·28-s − 22.6·29-s + 30.1·31-s + 237.·32-s + 329.·34-s − 130.·37-s + 507.·38-s + 347.·41-s − 26.7·43-s + 111.·44-s − 166.·46-s − 460.·47-s − 196.·49-s + 540.·52-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.38·4-s − 0.654·7-s − 0.602·8-s + 0.274·11-s + 1.03·13-s + 1.01·14-s − 0.458·16-s − 1.07·17-s − 1.40·19-s − 0.424·22-s + 0.345·23-s − 1.60·26-s − 0.909·28-s − 0.144·29-s + 0.174·31-s + 1.31·32-s + 1.66·34-s − 0.578·37-s + 2.16·38-s + 1.32·41-s − 0.0949·43-s + 0.381·44-s − 0.533·46-s − 1.43·47-s − 0.571·49-s + 1.44·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5719560476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5719560476\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.37T + 8T^{2} \) |
| 7 | \( 1 + 12.1T + 343T^{2} \) |
| 11 | \( 1 - 10.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 22.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 347.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 26.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 8.36T + 2.05e5T^{2} \) |
| 61 | \( 1 + 82.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 683.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 486.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 99.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 8.80T + 7.04e5T^{2} \) |
| 97 | \( 1 + 660.T + 9.12e5T^{2} \) |
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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
−8.723775388907963368157895885830, −8.388803382797914872596803430597, −7.34397465167248054743729491510, −6.56606916799766314166517330079, −6.13321551931219653972591991459, −4.66104450055483693054143952916, −3.71706231961849110151506649295, −2.49574185695734970553575731954, −1.56597719996452882934401440035, −0.44746934537890273910923762736,
0.44746934537890273910923762736, 1.56597719996452882934401440035, 2.49574185695734970553575731954, 3.71706231961849110151506649295, 4.66104450055483693054143952916, 6.13321551931219653972591991459, 6.56606916799766314166517330079, 7.34397465167248054743729491510, 8.388803382797914872596803430597, 8.723775388907963368157895885830
