L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 75.0·5-s + 36·6-s + 17.7·7-s − 64·8-s + 81·9-s + 300.·10-s + 485.·11-s − 144·12-s − 507.·13-s − 71.1·14-s + 675.·15-s + 256·16-s − 346.·17-s − 324·18-s − 1.66e3·19-s − 1.20e3·20-s − 160.·21-s − 1.94e3·22-s − 593.·23-s + 576·24-s + 2.50e3·25-s + 2.02e3·26-s − 729·27-s + 284.·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.34·5-s + 0.408·6-s + 0.137·7-s − 0.353·8-s + 0.333·9-s + 0.948·10-s + 1.21·11-s − 0.288·12-s − 0.832·13-s − 0.0970·14-s + 0.774·15-s + 0.250·16-s − 0.290·17-s − 0.235·18-s − 1.05·19-s − 0.671·20-s − 0.0792·21-s − 0.855·22-s − 0.234·23-s + 0.204·24-s + 0.801·25-s + 0.588·26-s − 0.192·27-s + 0.0686·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4551481436\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4551481436\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 5 | \( 1 + 75.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 17.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 485.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 507.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 346.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 593.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.84e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.59e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.69e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 5.68e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.55e4T + 8.58e9T^{2} \) |
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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
−10.91796658332606963771435347517, −9.635169403727546717507992527599, −8.812751257312267669651688454134, −7.71272378511999868179835053348, −7.08721318021821100552890550000, −6.00951071775992577087665269503, −4.53024788618163106151372581446, −3.65406285526802359757148546293, −1.88482348598650722023035153655, −0.41120633170087273669928793384,
0.41120633170087273669928793384, 1.88482348598650722023035153655, 3.65406285526802359757148546293, 4.53024788618163106151372581446, 6.00951071775992577087665269503, 7.08721318021821100552890550000, 7.71272378511999868179835053348, 8.812751257312267669651688454134, 9.635169403727546717507992527599, 10.91796658332606963771435347517