L(s) = 1 | − 3-s − 5-s − 2·9-s − 3·11-s − 13-s + 15-s + 5·17-s − 6·19-s + 25-s + 5·27-s − 5·29-s + 2·31-s + 3·33-s − 4·37-s + 39-s + 2·41-s − 10·43-s + 2·45-s − 9·47-s − 5·51-s + 6·53-s + 3·55-s + 6·57-s − 6·59-s + 12·61-s + 65-s + 2·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s + 1.21·17-s − 1.37·19-s + 1/5·25-s + 0.962·27-s − 0.928·29-s + 0.359·31-s + 0.522·33-s − 0.657·37-s + 0.160·39-s + 0.312·41-s − 1.52·43-s + 0.298·45-s − 1.31·47-s − 0.700·51-s + 0.824·53-s + 0.404·55-s + 0.794·57-s − 0.781·59-s + 1.53·61-s + 0.124·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7286189015\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7286189015\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−8.231230058258712000243092433127, −7.934819959822274081714681139985, −6.92288217725136468536000846238, −6.24242977831215165810735121848, −5.35318682894933643029769478154, −4.95702665735423568751724537401, −3.82239877858026214780958170553, −3.02250890666083408706942626208, −2.00393289721216725637791883713, −0.48300653155031577048533313748,
0.48300653155031577048533313748, 2.00393289721216725637791883713, 3.02250890666083408706942626208, 3.82239877858026214780958170553, 4.95702665735423568751724537401, 5.35318682894933643029769478154, 6.24242977831215165810735121848, 6.92288217725136468536000846238, 7.934819959822274081714681139985, 8.231230058258712000243092433127