L(s) = 1 | − 0.398·3-s + 1.24·5-s − 2.42·7-s − 2.84·9-s − 3.64·11-s + 0.544·13-s − 0.496·15-s + 5.31·17-s + 2.12·19-s + 0.966·21-s + 3.32·23-s − 3.45·25-s + 2.33·27-s + 4.85·29-s − 5.85·31-s + 1.45·33-s − 3.01·35-s − 7.60·37-s − 0.217·39-s + 7.84·41-s − 8.89·43-s − 3.53·45-s − 6.21·47-s − 1.13·49-s − 2.11·51-s + 6.99·53-s − 4.54·55-s + ⋯ |
L(s) = 1 | − 0.230·3-s + 0.556·5-s − 0.915·7-s − 0.946·9-s − 1.09·11-s + 0.151·13-s − 0.128·15-s + 1.28·17-s + 0.488·19-s + 0.210·21-s + 0.692·23-s − 0.690·25-s + 0.448·27-s + 0.902·29-s − 1.05·31-s + 0.253·33-s − 0.509·35-s − 1.24·37-s − 0.0347·39-s + 1.22·41-s − 1.35·43-s − 0.527·45-s − 0.905·47-s − 0.162·49-s − 0.296·51-s + 0.960·53-s − 0.612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.263938395\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.263938395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 + 0.398T + 3T^{2} \) |
| 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 - 0.544T + 13T^{2} \) |
| 17 | \( 1 - 5.31T + 17T^{2} \) |
| 19 | \( 1 - 2.12T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 5.85T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 6.99T + 53T^{2} \) |
| 59 | \( 1 - 0.330T + 59T^{2} \) |
| 61 | \( 1 - 0.464T + 61T^{2} \) |
| 67 | \( 1 - 8.75T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 + 0.404T + 89T^{2} \) |
| 97 | \( 1 + 1.71T + 97T^{2} \) |
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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
−8.409540342402438887004055866755, −7.76774969703324888702381404601, −6.88432978613656776251764094264, −6.13846830763031688530056834170, −5.43453870733475032069758233974, −5.06680074638077876591695842420, −3.51915670701762151785090212275, −3.07451546362427882201502883612, −2.07604967215811579201158134928, −0.62662386320975534789569874169,
0.62662386320975534789569874169, 2.07604967215811579201158134928, 3.07451546362427882201502883612, 3.51915670701762151785090212275, 5.06680074638077876591695842420, 5.43453870733475032069758233974, 6.13846830763031688530056834170, 6.88432978613656776251764094264, 7.76774969703324888702381404601, 8.409540342402438887004055866755