| L(s) = 1 | − 2-s + 1.87·3-s + 4-s − 1.87·6-s + 1.53·7-s − 8-s + 2.53·9-s + 1.87·12-s − 0.347·13-s − 1.53·14-s + 16-s + 0.347·17-s − 2.53·18-s − 19-s + 2.87·21-s − 1.87·23-s − 1.87·24-s + 0.347·26-s + 2.87·27-s + 1.53·28-s − 0.347·29-s − 32-s − 0.347·34-s + 2.53·36-s + 37-s + 38-s − 0.652·39-s + ⋯ |
| L(s) = 1 | − 2-s + 1.87·3-s + 4-s − 1.87·6-s + 1.53·7-s − 8-s + 2.53·9-s + 1.87·12-s − 0.347·13-s − 1.53·14-s + 16-s + 0.347·17-s − 2.53·18-s − 19-s + 2.87·21-s − 1.87·23-s − 1.87·24-s + 0.347·26-s + 2.87·27-s + 1.53·28-s − 0.347·29-s − 32-s − 0.347·34-s + 2.53·36-s + 37-s + 38-s − 0.652·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.910498097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910498097\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.87T + T^{2} \) |
| 7 | \( 1 - 1.53T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.347T + T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 23 | \( 1 + 1.87T + T^{2} \) |
| 29 | \( 1 + 0.347T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + 1.53T + T^{2} \) |
| 59 | \( 1 - 1.87T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−8.497247225194817449614646285560, −8.034035601476616957812394120734, −7.74696446579968495356410257820, −6.93300801448237260386046237234, −5.89163009425620121811726394799, −4.61408884470067770113623007625, −3.91515814680727749871202078643, −2.84304511510068920946380533510, −2.03319230706837636392119405231, −1.55059409854462001210166804268,
1.55059409854462001210166804268, 2.03319230706837636392119405231, 2.84304511510068920946380533510, 3.91515814680727749871202078643, 4.61408884470067770113623007625, 5.89163009425620121811726394799, 6.93300801448237260386046237234, 7.74696446579968495356410257820, 8.034035601476616957812394120734, 8.497247225194817449614646285560
