| L(s) = 1 | + 0.891·2-s − 0.205·4-s + 1.86·5-s − 1.07·8-s + 9-s + 1.66·10-s − 1.70·13-s − 0.752·16-s + 0.891·18-s − 0.547·19-s − 0.382·20-s + 2.47·25-s − 1.51·26-s + 0.184·29-s − 1.96·31-s + 0.403·32-s − 0.205·36-s − 0.487·38-s − 2.00·40-s + 0.184·43-s + 1.86·45-s + 49-s + 2.20·50-s + 0.349·52-s − 0.547·53-s + 0.164·58-s + 1.47·59-s + ⋯ |
| L(s) = 1 | + 0.891·2-s − 0.205·4-s + 1.86·5-s − 1.07·8-s + 9-s + 1.66·10-s − 1.70·13-s − 0.752·16-s + 0.891·18-s − 0.547·19-s − 0.382·20-s + 2.47·25-s − 1.51·26-s + 0.184·29-s − 1.96·31-s + 0.403·32-s − 0.205·36-s − 0.487·38-s − 2.00·40-s + 0.184·43-s + 1.86·45-s + 49-s + 2.20·50-s + 0.349·52-s − 0.547·53-s + 0.164·58-s + 1.47·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 991 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.750410265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.750410265\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 991 | \( 1 - T \) |
| good | 2 | \( 1 - 0.891T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.86T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.70T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.547T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 0.184T + T^{2} \) |
| 31 | \( 1 + 1.96T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.184T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.547T + T^{2} \) |
| 59 | \( 1 - 1.47T + T^{2} \) |
| 61 | \( 1 + 1.96T + T^{2} \) |
| 67 | \( 1 + 1.20T + T^{2} \) |
| 71 | \( 1 - 1.47T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.70T + T^{2} \) |
| 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
−9.996198953429350448626985854643, −9.520841069064073408658815152671, −8.866970107672524887904102008777, −7.37146280979417854443805044933, −6.57948566380600471636066284105, −5.64136966607014546243137790169, −5.06187702469306368279194211171, −4.20235094839647178606395042221, −2.77786894867274202422693280568, −1.85029238154871274051666353517,
1.85029238154871274051666353517, 2.77786894867274202422693280568, 4.20235094839647178606395042221, 5.06187702469306368279194211171, 5.64136966607014546243137790169, 6.57948566380600471636066284105, 7.37146280979417854443805044933, 8.866970107672524887904102008777, 9.520841069064073408658815152671, 9.996198953429350448626985854643
