| L(s) = 1 | − 1.24·2-s + 0.554·4-s + 1.97·5-s + 0.554·8-s − 2.46·10-s + 11-s − 1.24·16-s + 0.149·19-s + 1.09·20-s − 1.24·22-s − 1.91·23-s + 2.91·25-s + 1.46·29-s + 0.999·32-s − 0.186·38-s + 1.09·40-s + 0.445·41-s + 0.554·44-s + 2.38·46-s + 49-s − 3.63·50-s − 0.149·53-s + 1.97·55-s − 1.82·58-s − 0.730·59-s − 1.80·61-s + 0.0829·76-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.554·4-s + 1.97·5-s + 0.554·8-s − 2.46·10-s + 11-s − 1.24·16-s + 0.149·19-s + 1.09·20-s − 1.24·22-s − 1.91·23-s + 2.91·25-s + 1.46·29-s + 0.999·32-s − 0.186·38-s + 1.09·40-s + 0.445·41-s + 0.554·44-s + 2.38·46-s + 49-s − 3.63·50-s − 0.149·53-s + 1.97·55-s − 1.82·58-s − 0.730·59-s − 1.80·61-s + 0.0829·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.042655732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.042655732\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 431 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + T^{2} \) |
| 5 | \( 1 - 1.97T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.149T + T^{2} \) |
| 23 | \( 1 + 1.91T + T^{2} \) |
| 29 | \( 1 - 1.46T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.445T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.149T + T^{2} \) |
| 59 | \( 1 + 0.730T + T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 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 | \( 1 + 1.46T + T^{2} \) |
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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
−8.920755159919623405935925309041, −8.173298186009065670378446396135, −7.26075315580192662716665750045, −6.36865132478836235443473812951, −6.06010587904895189451466372442, −5.01302087650233792161109723965, −4.14770101540649931468450194611, −2.71178508903859154680448900016, −1.85649036724588609712676777829, −1.17791319844264995944488994243,
1.17791319844264995944488994243, 1.85649036724588609712676777829, 2.71178508903859154680448900016, 4.14770101540649931468450194611, 5.01302087650233792161109723965, 6.06010587904895189451466372442, 6.36865132478836235443473812951, 7.26075315580192662716665750045, 8.173298186009065670378446396135, 8.920755159919623405935925309041
