| L(s) = 1 | − 1.24·2-s + 0.554·4-s − 0.730·5-s + 0.554·8-s + 0.911·10-s + 11-s − 1.24·16-s + 1.65·19-s − 0.405·20-s − 1.24·22-s + 1.46·23-s − 0.466·25-s − 1.91·29-s + 0.999·32-s − 2.06·38-s − 0.405·40-s + 0.445·41-s + 0.554·44-s − 1.82·46-s + 49-s + 0.581·50-s − 1.65·53-s − 0.730·55-s + 2.38·58-s + 1.97·59-s − 1.80·61-s + 0.917·76-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.554·4-s − 0.730·5-s + 0.554·8-s + 0.911·10-s + 11-s − 1.24·16-s + 1.65·19-s − 0.405·20-s − 1.24·22-s + 1.46·23-s − 0.466·25-s − 1.91·29-s + 0.999·32-s − 2.06·38-s − 0.405·40-s + 0.445·41-s + 0.554·44-s − 1.82·46-s + 49-s + 0.581·50-s − 1.65·53-s − 0.730·55-s + 2.38·58-s + 1.97·59-s − 1.80·61-s + 0.917·76-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6004387971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6004387971\) |
| \(L(1)\) |
|
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 | \( 1 + 0.730T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.65T + T^{2} \) |
| 23 | \( 1 - 1.46T + T^{2} \) |
| 29 | \( 1 + 1.91T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.445T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.65T + T^{2} \) |
| 59 | \( 1 - 1.97T + T^{2} \) |
| 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 | \( 1 - 1.91T + T^{2} \) |
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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
−8.858692714593811880175643145939, −7.85773955172919230285935243159, −7.44420580533229391356215628287, −6.88257884818070948062900583641, −5.77545949611788223749129165852, −4.85587550521992039287306831805, −3.96013735450438196267619545790, −3.19641466382230086579357805905, −1.79589460852505334699791819281, −0.837185964523652171113799152016,
0.837185964523652171113799152016, 1.79589460852505334699791819281, 3.19641466382230086579357805905, 3.96013735450438196267619545790, 4.85587550521992039287306831805, 5.77545949611788223749129165852, 6.88257884818070948062900583641, 7.44420580533229391356215628287, 7.85773955172919230285935243159, 8.858692714593811880175643145939
