L(s) = 1 | + 2.74·2-s + 0.210·3-s + 5.53·4-s + 0.578·6-s + 2.32·7-s + 9.70·8-s − 2.95·9-s + 1.16·12-s + 0.534·13-s + 6.37·14-s + 15.5·16-s − 2.42·17-s − 8.11·18-s + 4.95·19-s + 0.489·21-s + 4.53·23-s + 2.04·24-s + 1.46·26-s − 1.25·27-s + 12.8·28-s − 5.48·29-s + 1.04·31-s + 23.3·32-s − 6.64·34-s − 16.3·36-s − 7.48·37-s + 13.6·38-s + ⋯ |
L(s) = 1 | + 1.94·2-s + 0.121·3-s + 2.76·4-s + 0.236·6-s + 0.878·7-s + 3.42·8-s − 0.985·9-s + 0.336·12-s + 0.148·13-s + 1.70·14-s + 3.88·16-s − 0.587·17-s − 1.91·18-s + 1.13·19-s + 0.106·21-s + 0.945·23-s + 0.417·24-s + 0.287·26-s − 0.241·27-s + 2.42·28-s − 1.01·29-s + 0.187·31-s + 4.11·32-s − 1.13·34-s − 2.72·36-s − 1.23·37-s + 2.20·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.517160046\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.517160046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 - 0.210T + 3T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 13 | \( 1 - 0.534T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 - 1.04T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 - 6.79T + 61T^{2} \) |
| 67 | \( 1 + 0.721T + 67T^{2} \) |
| 71 | \( 1 - 4.53T + 71T^{2} \) |
| 73 | \( 1 + 1.06T + 73T^{2} \) |
| 79 | \( 1 - 4.64T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 4.75T + 97T^{2} \) |
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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
−8.470669629390364564192420304721, −7.69242694742979878708724456499, −7.00911907499464797956598943991, −6.18407972478353394771309251226, −5.32866221953114655951117733982, −5.04392618434504263676808750533, −4.03455609142391422578866989938, −3.24368319781624249145716824969, −2.48398123493679684078554270968, −1.47839825682678563261693247171,
1.47839825682678563261693247171, 2.48398123493679684078554270968, 3.24368319781624249145716824969, 4.03455609142391422578866989938, 5.04392618434504263676808750533, 5.32866221953114655951117733982, 6.18407972478353394771309251226, 7.00911907499464797956598943991, 7.69242694742979878708724456499, 8.470669629390364564192420304721