| L(s) = 1 | − 1.35·2-s + 0.00323·3-s − 0.163·4-s − 0.00438·6-s − 1.75·7-s + 2.93·8-s − 2.99·9-s + 0.520·11-s − 0.000527·12-s − 6.40·13-s + 2.37·14-s − 3.64·16-s + 7.23·17-s + 4.06·18-s + 7.74·19-s − 0.00566·21-s − 0.705·22-s + 4.04·23-s + 0.00948·24-s + 8.68·26-s − 0.0194·27-s + 0.285·28-s − 7.52·29-s − 0.488·31-s − 0.920·32-s + 0.00168·33-s − 9.80·34-s + ⋯ |
| L(s) = 1 | − 0.958·2-s + 0.00186·3-s − 0.0815·4-s − 0.00178·6-s − 0.662·7-s + 1.03·8-s − 0.999·9-s + 0.156·11-s − 0.000152·12-s − 1.77·13-s + 0.634·14-s − 0.911·16-s + 1.75·17-s + 0.958·18-s + 1.77·19-s − 0.00123·21-s − 0.150·22-s + 0.843·23-s + 0.00193·24-s + 1.70·26-s − 0.00373·27-s + 0.0540·28-s − 1.39·29-s − 0.0877·31-s − 0.162·32-s + 0.000293·33-s − 1.68·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5989240105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5989240105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 0.00323T + 3T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 - 0.520T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 7.23T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 - 4.04T + 23T^{2} \) |
| 29 | \( 1 + 7.52T + 29T^{2} \) |
| 31 | \( 1 + 0.488T + 31T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 + 0.613T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 + 4.01T + 67T^{2} \) |
| 71 | \( 1 - 3.35T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 0.873T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.88617664243805591051528644621, −7.63357408518245558430588028660, −7.02079145738898798827296898608, −5.83902984495367586854612810604, −5.30827552947685503399102989548, −4.58443010919246702116632773300, −3.29519082129241806878291191601, −2.90303228034557680551439906938, −1.55500858252755525290404855216, −0.48490724493237970586258630807,
0.48490724493237970586258630807, 1.55500858252755525290404855216, 2.90303228034557680551439906938, 3.29519082129241806878291191601, 4.58443010919246702116632773300, 5.30827552947685503399102989548, 5.83902984495367586854612810604, 7.02079145738898798827296898608, 7.63357408518245558430588028660, 7.88617664243805591051528644621
