L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 6·11-s − 12-s − 13-s + 14-s + 16-s − 18-s − 4·19-s + 21-s + 6·22-s + 3·23-s + 24-s + 26-s − 27-s − 28-s − 3·29-s − 5·31-s − 32-s + 6·33-s + 36-s + 10·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 1.27·22-s + 0.625·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 1.04·33-s + 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3998162242\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3998162242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.65001847443107, −12.27273692721606, −11.66293878539781, −11.04678372794323, −10.88629619329869, −10.41012621762849, −10.00877838080591, −9.491282352237031, −9.128156286476885, −8.389292697827047, −8.095289901300170, −7.586337608311205, −7.127343335442131, −6.651201085834693, −6.180420522487135, −5.503960460086181, −5.268920186900588, −4.718458067737247, −3.996325118204422, −3.443831475492084, −2.696879109907702, −2.359869922945168, −1.755851801107971, −0.8690387221822015, −0.2392953185551831,
0.2392953185551831, 0.8690387221822015, 1.755851801107971, 2.359869922945168, 2.696879109907702, 3.443831475492084, 3.996325118204422, 4.718458067737247, 5.268920186900588, 5.503960460086181, 6.180420522487135, 6.651201085834693, 7.127343335442131, 7.586337608311205, 8.095289901300170, 8.389292697827047, 9.128156286476885, 9.491282352237031, 10.00877838080591, 10.41012621762849, 10.88629619329869, 11.04678372794323, 11.66293878539781, 12.27273692721606, 12.65001847443107