L(s) = 1 | − 0.268·3-s + 5.16·5-s − 7·7-s − 26.9·9-s + 11·11-s + 74.5·13-s − 1.38·15-s + 53.1·17-s − 133.·19-s + 1.88·21-s + 123.·23-s − 98.2·25-s + 14.4·27-s + 36.7·29-s − 122.·31-s − 2.95·33-s − 36.1·35-s + 161.·37-s − 20.0·39-s + 299.·41-s − 467.·43-s − 139.·45-s + 1.35·47-s + 49·49-s − 14.2·51-s + 399.·53-s + 56.8·55-s + ⋯ |
L(s) = 1 | − 0.0517·3-s + 0.462·5-s − 0.377·7-s − 0.997·9-s + 0.301·11-s + 1.59·13-s − 0.0239·15-s + 0.758·17-s − 1.61·19-s + 0.0195·21-s + 1.11·23-s − 0.786·25-s + 0.103·27-s + 0.235·29-s − 0.712·31-s − 0.0155·33-s − 0.174·35-s + 0.715·37-s − 0.0822·39-s + 1.14·41-s − 1.65·43-s − 0.461·45-s + 0.00420·47-s + 0.142·49-s − 0.0392·51-s + 1.03·53-s + 0.139·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.048543159\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.048543159\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
good | 3 | \( 1 + 0.268T + 27T^{2} \) |
| 5 | \( 1 - 5.16T + 125T^{2} \) |
| 13 | \( 1 - 74.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 36.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 1.35T + 1.03e5T^{2} \) |
| 53 | \( 1 - 399.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 215.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 361.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 148.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 634.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 538.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 931.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 31.6T + 9.12e5T^{2} \) |
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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
−9.166655273256739144083320057061, −8.678078293269759289252187609373, −7.85109367676687452669377103951, −6.55360489702276124604827611931, −6.10405682231156796122045254499, −5.29880549052877430805133008488, −3.99756377400622439894318898545, −3.17787610944677170850330360455, −2.00641204047008128670901647266, −0.73245876370529098184909633495,
0.73245876370529098184909633495, 2.00641204047008128670901647266, 3.17787610944677170850330360455, 3.99756377400622439894318898545, 5.29880549052877430805133008488, 6.10405682231156796122045254499, 6.55360489702276124604827611931, 7.85109367676687452669377103951, 8.678078293269759289252187609373, 9.166655273256739144083320057061