L(s) = 1 | − 0.801·2-s − 1.35·4-s + 2.44·5-s − 7-s + 2.69·8-s − 1.96·10-s − 1.49·11-s + 2.93·13-s + 0.801·14-s + 0.554·16-s + 5.93·17-s − 3.02·19-s − 3.31·20-s + 1.19·22-s − 4.74·23-s + 0.978·25-s − 2.35·26-s + 1.35·28-s + 9.25·29-s − 5.18·31-s − 5.82·32-s − 4.76·34-s − 2.44·35-s + 1.06·37-s + 2.42·38-s + 6.58·40-s + 0.902·41-s + ⋯ |
L(s) = 1 | − 0.567·2-s − 0.678·4-s + 1.09·5-s − 0.377·7-s + 0.951·8-s − 0.620·10-s − 0.450·11-s + 0.815·13-s + 0.214·14-s + 0.138·16-s + 1.44·17-s − 0.694·19-s − 0.741·20-s + 0.255·22-s − 0.988·23-s + 0.195·25-s − 0.462·26-s + 0.256·28-s + 1.71·29-s − 0.931·31-s − 1.03·32-s − 0.816·34-s − 0.413·35-s + 0.174·37-s + 0.393·38-s + 1.04·40-s + 0.140·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1359 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.237634516\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.237634516\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 151 | \( 1 + T \) |
good | 2 | \( 1 + 0.801T + 2T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 - 5.93T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 9.25T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 - 0.902T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 1.91T + 59T^{2} \) |
| 61 | \( 1 - 7.00T + 61T^{2} \) |
| 67 | \( 1 - 0.241T + 67T^{2} \) |
| 71 | \( 1 + 7.64T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 - 4.38T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−9.617940502359472836474013182443, −8.885438984236031433826723432828, −8.172318796399403855194795048431, −7.34467645643644588633940258701, −6.07472634257446465983815934172, −5.66287371613571262203160692241, −4.52645621668879130941180424921, −3.48943459510460487041566243613, −2.15929289786023689856585821236, −0.920766014738660367430034789557,
0.920766014738660367430034789557, 2.15929289786023689856585821236, 3.48943459510460487041566243613, 4.52645621668879130941180424921, 5.66287371613571262203160692241, 6.07472634257446465983815934172, 7.34467645643644588633940258701, 8.172318796399403855194795048431, 8.885438984236031433826723432828, 9.617940502359472836474013182443