| L(s) = 1 | + 2.56·2-s − 3·3-s − 1.43·4-s − 7.68·6-s − 6.24·7-s − 24.1·8-s + 9·9-s − 11·11-s + 4.31·12-s + 49.1·13-s − 16·14-s − 50.4·16-s − 82.7·17-s + 23.0·18-s − 130.·19-s + 18.7·21-s − 28.1·22-s + 185.·23-s + 72.5·24-s + 125.·26-s − 27·27-s + 8.98·28-s − 8.90·29-s + 5.26·31-s + 64.2·32-s + 33·33-s − 211.·34-s + ⋯ |
| L(s) = 1 | + 0.905·2-s − 0.577·3-s − 0.179·4-s − 0.522·6-s − 0.337·7-s − 1.06·8-s + 0.333·9-s − 0.301·11-s + 0.103·12-s + 1.04·13-s − 0.305·14-s − 0.787·16-s − 1.17·17-s + 0.301·18-s − 1.57·19-s + 0.194·21-s − 0.273·22-s + 1.68·23-s + 0.616·24-s + 0.949·26-s − 0.192·27-s + 0.0606·28-s − 0.0570·29-s + 0.0304·31-s + 0.354·32-s + 0.174·33-s − 1.06·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.697234242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697234242\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.56T + 8T^{2} \) |
| 7 | \( 1 + 6.24T + 343T^{2} \) |
| 13 | \( 1 - 49.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 130.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 8.90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.26T + 2.97e4T^{2} \) |
| 37 | \( 1 - 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 513.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 557.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 168.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 786.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 339.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 309.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 798.T + 9.12e5T^{2} \) |
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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
−9.876528510536384947953267090958, −8.954080081240248141494269535549, −8.277786429051531621205711879598, −6.71788831850485457084068465706, −6.32318897000562416886075608313, −5.28967833031309285134040649406, −4.47846261270584869041862970750, −3.66192660395499297026734709011, −2.42307071631622425828895461978, −0.63797008900176234647170914173,
0.63797008900176234647170914173, 2.42307071631622425828895461978, 3.66192660395499297026734709011, 4.47846261270584869041862970750, 5.28967833031309285134040649406, 6.32318897000562416886075608313, 6.71788831850485457084068465706, 8.277786429051531621205711879598, 8.954080081240248141494269535549, 9.876528510536384947953267090958
