| L(s) = 1 | − 2-s − 3-s − 4-s + 3.85·5-s + 6-s + 7-s + 3·8-s + 9-s − 3.85·10-s + 12-s + 3.23·13-s − 14-s − 3.85·15-s − 16-s + 1.14·17-s − 18-s + 0.381·19-s − 3.85·20-s − 21-s + 8.09·23-s − 3·24-s + 9.85·25-s − 3.23·26-s − 27-s − 28-s − 2·29-s + 3.85·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.5·4-s + 1.72·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 0.333·9-s − 1.21·10-s + 0.288·12-s + 0.897·13-s − 0.267·14-s − 0.995·15-s − 0.250·16-s + 0.277·17-s − 0.235·18-s + 0.0876·19-s − 0.861·20-s − 0.218·21-s + 1.68·23-s − 0.612·24-s + 1.97·25-s − 0.634·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.703·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496644372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496644372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + 2T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 - 0.381T + 19T^{2} \) |
| 23 | \( 1 - 8.09T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 + 0.909T + 37T^{2} \) |
| 41 | \( 1 + 4.14T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 0.763T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.912033009596170760613377033710, −8.478633419871655741724996792695, −7.35960602672477971935752710109, −6.55769948134251782445651093260, −5.74724876495365417589612048001, −5.14786788886546983980070557020, −4.37549129188365620247253202142, −2.96521192891050277510016640714, −1.64436112031629151714023944719, −1.01115833000420976867150913795,
1.01115833000420976867150913795, 1.64436112031629151714023944719, 2.96521192891050277510016640714, 4.37549129188365620247253202142, 5.14786788886546983980070557020, 5.74724876495365417589612048001, 6.55769948134251782445651093260, 7.35960602672477971935752710109, 8.478633419871655741724996792695, 8.912033009596170760613377033710
