L(s) = 1 | − 0.974·3-s + 0.130·5-s − 2.76·7-s − 2.05·9-s + 3.51·11-s − 0.127·15-s + 3.67·17-s − 4.63·19-s + 2.69·21-s + 0.213·23-s − 4.98·25-s + 4.92·27-s + 9.93·29-s − 5.21·31-s − 3.42·33-s − 0.361·35-s + 2.42·37-s + 11.1·41-s + 7.00·43-s − 0.268·45-s + 6.20·47-s + 0.646·49-s − 3.57·51-s + 6.51·53-s + 0.459·55-s + 4.51·57-s − 7.31·59-s + ⋯ |
L(s) = 1 | − 0.562·3-s + 0.0584·5-s − 1.04·7-s − 0.683·9-s + 1.05·11-s − 0.0329·15-s + 0.890·17-s − 1.06·19-s + 0.588·21-s + 0.0445·23-s − 0.996·25-s + 0.947·27-s + 1.84·29-s − 0.937·31-s − 0.595·33-s − 0.0611·35-s + 0.399·37-s + 1.74·41-s + 1.06·43-s − 0.0399·45-s + 0.905·47-s + 0.0923·49-s − 0.501·51-s + 0.895·53-s + 0.0619·55-s + 0.598·57-s − 0.952·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.091930431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.091930431\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.974T + 3T^{2} \) |
| 5 | \( 1 - 0.130T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 3.51T + 11T^{2} \) |
| 17 | \( 1 - 3.67T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 - 0.213T + 23T^{2} \) |
| 29 | \( 1 - 9.93T + 29T^{2} \) |
| 31 | \( 1 + 5.21T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 7.00T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 - 6.51T + 53T^{2} \) |
| 59 | \( 1 + 7.31T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 7.85T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 + 0.767T + 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−9.543106605741053235870752295559, −8.969588759384931020205214028611, −8.010019040449092895083577956987, −6.94770213968353698331116076089, −6.14782441658861282003779482416, −5.76885607433801811174188034314, −4.44440193039723824653612282760, −3.54608018428676937152877656033, −2.46684707841988724330257037278, −0.77236381406630870503032624695,
0.77236381406630870503032624695, 2.46684707841988724330257037278, 3.54608018428676937152877656033, 4.44440193039723824653612282760, 5.76885607433801811174188034314, 6.14782441658861282003779482416, 6.94770213968353698331116076089, 8.010019040449092895083577956987, 8.969588759384931020205214028611, 9.543106605741053235870752295559