| L(s) = 1 | + 1.53·3-s − 2.62·5-s − 2.98·7-s − 0.651·9-s − 1.73·11-s − 1.44·13-s − 4.02·15-s + 5.79·17-s + 1.23·19-s − 4.56·21-s − 0.489·23-s + 1.89·25-s − 5.59·27-s − 10.2·29-s − 8.27·31-s − 2.65·33-s + 7.82·35-s − 10.1·37-s − 2.21·39-s + 11.1·41-s + 9.61·43-s + 1.70·45-s + 6.33·47-s + 1.88·49-s + 8.88·51-s + 5.25·53-s + 4.54·55-s + ⋯ |
| L(s) = 1 | + 0.884·3-s − 1.17·5-s − 1.12·7-s − 0.217·9-s − 0.522·11-s − 0.401·13-s − 1.03·15-s + 1.40·17-s + 0.283·19-s − 0.996·21-s − 0.102·23-s + 0.378·25-s − 1.07·27-s − 1.90·29-s − 1.48·31-s − 0.462·33-s + 1.32·35-s − 1.66·37-s − 0.354·39-s + 1.73·41-s + 1.46·43-s + 0.254·45-s + 0.924·47-s + 0.269·49-s + 1.24·51-s + 0.721·53-s + 0.613·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.043088586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043088586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 0.489T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 8.27T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 - 5.25T + 53T^{2} \) |
| 59 | \( 1 - 2.12T + 59T^{2} \) |
| 61 | \( 1 + 0.448T + 61T^{2} \) |
| 67 | \( 1 - 5.39T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 2.28T + 89T^{2} \) |
| 97 | \( 1 - 2.14T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87750610405776618476873668935, −7.55184873637348296588316627289, −7.03054857317390046927103164125, −5.72242930613530215716547719914, −5.42957195277573533442600885425, −3.94732645329223881521972766876, −3.65745170381635938470936498458, −2.98090792407877179452462995072, −2.11284991864930883232085978439, −0.48273498846807278864046939035,
0.48273498846807278864046939035, 2.11284991864930883232085978439, 2.98090792407877179452462995072, 3.65745170381635938470936498458, 3.94732645329223881521972766876, 5.42957195277573533442600885425, 5.72242930613530215716547719914, 7.03054857317390046927103164125, 7.55184873637348296588316627289, 7.87750610405776618476873668935
