| L(s) = 1 | + 1.24·2-s − 1.80·3-s + 0.554·4-s − 0.445·5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s − 0.554·10-s − 0.999·12-s + 0.801·15-s − 1.24·16-s + 2.80·18-s + 1.24·19-s − 0.246·20-s + 1.00·24-s − 0.801·25-s − 2.24·27-s − 1.80·29-s + 0.999·30-s − 0.999·32-s + 1.24·36-s + 1.24·37-s + 1.55·38-s + 0.246·40-s − 0.445·43-s − 45-s + 2.24·48-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 1.80·3-s + 0.554·4-s − 0.445·5-s − 2.24·6-s − 0.554·8-s + 2.24·9-s − 0.554·10-s − 0.999·12-s + 0.801·15-s − 1.24·16-s + 2.80·18-s + 1.24·19-s − 0.246·20-s + 1.00·24-s − 0.801·25-s − 2.24·27-s − 1.80·29-s + 0.999·30-s − 0.999·32-s + 1.24·36-s + 1.24·37-s + 1.55·38-s + 0.246·40-s − 0.445·43-s − 45-s + 2.24·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5214110656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5214110656\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| good | 2 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 + 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.80T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - 1.24T + T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−15.04069640003003501555800810731, −13.58399519906890803499351778176, −12.62842911673974380337261507666, −11.75493220562099759167484344236, −11.17081547941659221637655464575, −9.646341880691708918339787590058, −7.34866153009380292102541584258, −6.02203245587172561884189386059, −5.19361966854786324344583978535, −3.95531641366854447656333489742,
3.95531641366854447656333489742, 5.19361966854786324344583978535, 6.02203245587172561884189386059, 7.34866153009380292102541584258, 9.646341880691708918339787590058, 11.17081547941659221637655464575, 11.75493220562099759167484344236, 12.62842911673974380337261507666, 13.58399519906890803499351778176, 15.04069640003003501555800810731
