| L(s) = 1 | + 1.24·2-s + 0.554·4-s − 1.80·5-s − 0.554·8-s + 9-s − 2.24·10-s − 0.445·11-s − 1.24·16-s − 0.445·17-s + 1.24·18-s + 1.24·19-s − 0.999·20-s − 0.554·22-s + 2.24·25-s + 1.24·29-s − 1.80·31-s − 0.999·32-s − 0.554·34-s + 0.554·36-s − 0.445·37-s + 1.55·38-s + 1.00·40-s − 1.80·43-s − 0.246·44-s − 1.80·45-s + 1.24·47-s + 49-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.554·4-s − 1.80·5-s − 0.554·8-s + 9-s − 2.24·10-s − 0.445·11-s − 1.24·16-s − 0.445·17-s + 1.24·18-s + 1.24·19-s − 0.999·20-s − 0.554·22-s + 2.24·25-s + 1.24·29-s − 1.80·31-s − 0.999·32-s − 0.554·34-s + 0.554·36-s − 0.445·37-s + 1.55·38-s + 1.00·40-s − 1.80·43-s − 0.246·44-s − 1.80·45-s + 1.24·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8764430973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8764430973\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 - T \) |
| good | 2 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 + 1.80T + T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - 1.24T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−13.13772808388056326616289293457, −12.29973644682524153420696357185, −11.70141259123624616684364305266, −10.59481161442468845621401137610, −9.013192558661001383822127103355, −7.72172489061707095845989243299, −6.86724953513348673953115213883, −5.13978808114584291835441902554, −4.20805702542999305543125531760, −3.27308661500225015771481383375,
3.27308661500225015771481383375, 4.20805702542999305543125531760, 5.13978808114584291835441902554, 6.86724953513348673953115213883, 7.72172489061707095845989243299, 9.013192558661001383822127103355, 10.59481161442468845621401137610, 11.70141259123624616684364305266, 12.29973644682524153420696357185, 13.13772808388056326616289293457
