L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s − 4·13-s + 16-s − 3·17-s − 18-s − 19-s + 2·22-s − 3·23-s + 24-s + 4·26-s − 27-s − 6·29-s + 3·31-s − 32-s + 2·33-s + 3·34-s + 36-s + 8·37-s + 38-s + 4·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.229·19-s + 0.426·22-s − 0.625·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 1.11·29-s + 0.538·31-s − 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s + 1.31·37-s + 0.162·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5899431158\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5899431158\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.20576376547836, −12.77064544221089, −12.52812270739446, −11.87382056044624, −11.28943901050240, −11.09677553927128, −10.54662194234655, −9.984331661372618, −9.535461397667327, −9.283633477194050, −8.516220218915895, −7.823355459157492, −7.750074379581554, −7.045713884642333, −6.543208328781398, −6.050267426521497, −5.439618508066291, −5.039725956688195, −4.193286401235639, −3.995773837134452, −2.814071606092355, −2.479473548611151, −1.915889208808026, −1.007238015287822, −0.3034177428105993,
0.3034177428105993, 1.007238015287822, 1.915889208808026, 2.479473548611151, 2.814071606092355, 3.995773837134452, 4.193286401235639, 5.039725956688195, 5.439618508066291, 6.050267426521497, 6.543208328781398, 7.045713884642333, 7.750074379581554, 7.823355459157492, 8.516220218915895, 9.283633477194050, 9.535461397667327, 9.984331661372618, 10.54662194234655, 11.09677553927128, 11.28943901050240, 11.87382056044624, 12.52812270739446, 12.77064544221089, 13.20576376547836