| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 17.0·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 34.1·10-s − 61.9·11-s − 12·12-s + 13·13-s − 14·14-s + 51.2·15-s + 16·16-s + 36.7·17-s + 18·18-s + 102.·19-s − 68.2·20-s + 21·21-s − 123.·22-s − 197.·23-s − 24·24-s + 166.·25-s + 26·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.07·10-s − 1.69·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.881·15-s + 0.250·16-s + 0.524·17-s + 0.235·18-s + 1.23·19-s − 0.763·20-s + 0.218·21-s − 1.20·22-s − 1.78·23-s − 0.204·24-s + 1.33·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.345087210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345087210\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 17.0T + 125T^{2} \) |
| 11 | \( 1 + 61.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 36.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 249.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 135.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 22.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 136.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 800.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 994.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 526.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 90.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 205.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−10.57298547991222518096872598585, −9.870765428879959608669923372406, −8.022987045468692264436945964334, −7.86259758057500287673712860966, −6.68155087205438743337154092528, −5.60811099853644938974540731664, −4.72368020022545884117791332098, −3.74126971840602204429961587063, −2.74675040156860044874362254419, −0.63470299444771382114383411091,
0.63470299444771382114383411091, 2.74675040156860044874362254419, 3.74126971840602204429961587063, 4.72368020022545884117791332098, 5.60811099853644938974540731664, 6.68155087205438743337154092528, 7.86259758057500287673712860966, 8.022987045468692264436945964334, 9.870765428879959608669923372406, 10.57298547991222518096872598585
