| L(s) = 1 | − 5.15·2-s + 18.5·4-s − 17.2·7-s − 54.3·8-s − 11·11-s − 3.57·13-s + 88.7·14-s + 131.·16-s − 101.·17-s + 87.5·19-s + 56.6·22-s − 123.·23-s + 18.4·26-s − 319.·28-s + 26.9·29-s − 174.·31-s − 244.·32-s + 524.·34-s + 329.·37-s − 451.·38-s − 21.1·41-s + 363.·43-s − 204.·44-s + 635.·46-s − 334.·47-s − 46.6·49-s − 66.2·52-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.31·4-s − 0.929·7-s − 2.40·8-s − 0.301·11-s − 0.0762·13-s + 1.69·14-s + 2.06·16-s − 1.45·17-s + 1.05·19-s + 0.549·22-s − 1.11·23-s + 0.138·26-s − 2.15·28-s + 0.172·29-s − 1.01·31-s − 1.35·32-s + 2.64·34-s + 1.46·37-s − 1.92·38-s − 0.0804·41-s + 1.29·43-s − 0.699·44-s + 2.03·46-s − 1.03·47-s − 0.136·49-s − 0.176·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2485392210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2485392210\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.15T + 8T^{2} \) |
| 7 | \( 1 + 17.2T + 343T^{2} \) |
| 13 | \( 1 + 3.57T + 2.19e3T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 26.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 329.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 21.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 649.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 755.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 469.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 508.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 213.T + 9.12e5T^{2} \) |
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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
−8.755821080543845617652199216069, −7.79345165921093397928747331882, −7.42486411778108264029630385032, −6.39340632929659304234669867011, −6.04755913443394061313945153587, −4.60919912455151256307880561544, −3.29907756595050987226189553866, −2.47858540082568906178487329847, −1.52788378749945216592204508172, −0.28812462431259916711654908669,
0.28812462431259916711654908669, 1.52788378749945216592204508172, 2.47858540082568906178487329847, 3.29907756595050987226189553866, 4.60919912455151256307880561544, 6.04755913443394061313945153587, 6.39340632929659304234669867011, 7.42486411778108264029630385032, 7.79345165921093397928747331882, 8.755821080543845617652199216069
