| L(s) = 1 | + 2·2-s + 0.111·3-s + 4·4-s + 11.3·5-s + 0.223·6-s + 3.28·7-s + 8·8-s − 26.9·9-s + 22.6·10-s + 48.3·11-s + 0.447·12-s + 21.9·13-s + 6.57·14-s + 1.26·15-s + 16·16-s + 19.9·17-s − 53.9·18-s − 116.·19-s + 45.3·20-s + 0.367·21-s + 96.6·22-s − 191.·23-s + 0.894·24-s + 3.41·25-s + 43.8·26-s − 6.03·27-s + 13.1·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0215·3-s + 0.5·4-s + 1.01·5-s + 0.0152·6-s + 0.177·7-s + 0.353·8-s − 0.999·9-s + 0.716·10-s + 1.32·11-s + 0.0107·12-s + 0.467·13-s + 0.125·14-s + 0.0218·15-s + 0.250·16-s + 0.284·17-s − 0.706·18-s − 1.40·19-s + 0.506·20-s + 0.00382·21-s + 0.936·22-s − 1.73·23-s + 0.00761·24-s + 0.0272·25-s + 0.330·26-s − 0.0430·27-s + 0.0887·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.470789581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.470789581\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 37 | \( 1 - 37T \) |
| good | 3 | \( 1 - 0.111T + 27T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 - 3.28T + 343T^{2} \) |
| 11 | \( 1 - 48.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 32.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 369.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 230.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 49.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 907.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 67.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 661.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 30.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 709.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 207.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.28819114742831573830677955516, −13.14378490818045289560446345440, −11.96303972778120044345672273628, −10.94722959869904705819470244722, −9.592356125945876713254049438119, −8.350406552697053301663673365327, −6.45588116433919486410873472989, −5.70214596633901270947942535338, −3.92588239213037693408896524305, −2.02630962582907765948964943083,
2.02630962582907765948964943083, 3.92588239213037693408896524305, 5.70214596633901270947942535338, 6.45588116433919486410873472989, 8.350406552697053301663673365327, 9.592356125945876713254049438119, 10.94722959869904705819470244722, 11.96303972778120044345672273628, 13.14378490818045289560446345440, 14.28819114742831573830677955516
