L(s) = 1 | − 0.191·3-s − 3.65·5-s + 2.84·7-s − 2.96·9-s + 11-s + 2.64·13-s + 0.699·15-s + 3.79·17-s − 0.211·19-s − 0.544·21-s + 6.37·23-s + 8.39·25-s + 1.13·27-s − 1.95·29-s − 2.76·31-s − 0.191·33-s − 10.4·35-s − 5.91·37-s − 0.505·39-s − 9.71·41-s − 7.53·43-s + 10.8·45-s + 12.2·47-s + 1.10·49-s − 0.724·51-s + 5.38·53-s − 3.65·55-s + ⋯ |
L(s) = 1 | − 0.110·3-s − 1.63·5-s + 1.07·7-s − 0.987·9-s + 0.301·11-s + 0.733·13-s + 0.180·15-s + 0.919·17-s − 0.0484·19-s − 0.118·21-s + 1.32·23-s + 1.67·25-s + 0.219·27-s − 0.362·29-s − 0.496·31-s − 0.0332·33-s − 1.76·35-s − 0.971·37-s − 0.0809·39-s − 1.51·41-s − 1.14·43-s + 1.61·45-s + 1.79·47-s + 0.157·49-s − 0.101·51-s + 0.739·53-s − 0.493·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.350992309\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350992309\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
good | 3 | \( 1 + 0.191T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 + 0.211T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 + 7.53T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 - 0.674T + 59T^{2} \) |
| 61 | \( 1 + 5.33T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 3.49T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 - 0.712T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 1.85T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−8.157910221146556992024703838924, −7.42742288620555263671033946585, −6.91064153192482836444702575255, −5.79765795043890591230850118946, −5.14044571842389800950746426330, −4.45216541458891914909775306988, −3.53564229344318466486458217743, −3.13351725088033820152493113465, −1.68076607753610580826015843337, −0.63374372581958260733893837284,
0.63374372581958260733893837284, 1.68076607753610580826015843337, 3.13351725088033820152493113465, 3.53564229344318466486458217743, 4.45216541458891914909775306988, 5.14044571842389800950746426330, 5.79765795043890591230850118946, 6.91064153192482836444702575255, 7.42742288620555263671033946585, 8.157910221146556992024703838924