| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 14.1·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 28.2·10-s − 20.4·11-s − 12·12-s − 13·13-s + 14·14-s + 42.4·15-s + 16·16-s − 82.1·17-s + 18·18-s + 96.4·19-s − 56.5·20-s − 21·21-s − 40.8·22-s + 184.·23-s − 24·24-s + 74.8·25-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 − 1.26·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.894·10-s − 0.559·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.730·15-s + 0.250·16-s − 1.17·17-s + 0.235·18-s + 1.16·19-s − 0.632·20-s − 0.218·21-s − 0.395·22-s + 1.67·23-s − 0.204·24-s + 0.599·25-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
| \(L(2)\) |
\(\approx\) |
\(1.861944640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861944640\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 14.1T + 125T^{2} \) |
| 11 | \( 1 + 20.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 82.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 184.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 34.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 33.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 27.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 313.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 179.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 187.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 181.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 430.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 910.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 195.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 176.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 95.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 923.T + 9.12e5T^{2} \) |
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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
−11.05808232913664764457629619636, −9.621636945726098002445078086362, −8.395323291167980654762660740154, −7.45402868384674430288403239055, −6.86794663428089351509660255608, −5.49836972508644200486616307014, −4.74171215150132614890961168052, −3.86062720984880498343155847378, −2.60698312010280308631417270104, −0.76757527469843736963342837434,
0.76757527469843736963342837434, 2.60698312010280308631417270104, 3.86062720984880498343155847378, 4.74171215150132614890961168052, 5.49836972508644200486616307014, 6.86794663428089351509660255608, 7.45402868384674430288403239055, 8.395323291167980654762660740154, 9.621636945726098002445078086362, 11.05808232913664764457629619636
