L(s) = 1 | − 0.858·2-s − 1.26·4-s − 2.26·5-s − 1.17·7-s + 2.80·8-s + 1.94·10-s + 2.14·11-s − 2.70·13-s + 1.01·14-s + 0.121·16-s + 4.21·17-s + 19-s + 2.85·20-s − 1.83·22-s + 3.95·23-s + 0.121·25-s + 2.32·26-s + 1.48·28-s − 4.38·29-s + 0.583·31-s − 5.70·32-s − 3.62·34-s + 2.66·35-s + 0.942·37-s − 0.858·38-s − 6.33·40-s + 9.93·41-s + ⋯ |
L(s) = 1 | − 0.607·2-s − 0.631·4-s − 1.01·5-s − 0.445·7-s + 0.990·8-s + 0.614·10-s + 0.645·11-s − 0.750·13-s + 0.270·14-s + 0.0303·16-s + 1.02·17-s + 0.229·19-s + 0.639·20-s − 0.391·22-s + 0.824·23-s + 0.0243·25-s + 0.455·26-s + 0.281·28-s − 0.814·29-s + 0.104·31-s − 1.00·32-s − 0.620·34-s + 0.450·35-s + 0.154·37-s − 0.139·38-s − 1.00·40-s + 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6926449294\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6926449294\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 0.858T + 2T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 - 2.14T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 0.583T + 31T^{2} \) |
| 37 | \( 1 - 0.942T + 37T^{2} \) |
| 41 | \( 1 - 9.93T + 41T^{2} \) |
| 43 | \( 1 + 8.96T + 43T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 6.86T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 0.243T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 0.743T + 97T^{2} \) |
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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
−7.929711226286837963636708137864, −7.29884459517697819067740311382, −6.76093326035721114928231086017, −5.62148208791467012572207395526, −5.02662766183994758375677112819, −4.09552754339336061393970108111, −3.70377492747342575910878394235, −2.75869794828450214439917149644, −1.41613648226481619908544638826, −0.49282616931618705194788419844,
0.49282616931618705194788419844, 1.41613648226481619908544638826, 2.75869794828450214439917149644, 3.70377492747342575910878394235, 4.09552754339336061393970108111, 5.02662766183994758375677112819, 5.62148208791467012572207395526, 6.76093326035721114928231086017, 7.29884459517697819067740311382, 7.929711226286837963636708137864