L(s) = 1 | + 2.12·2-s − 3-s + 2.53·4-s − 2.65·5-s − 2.12·6-s + 7-s + 1.12·8-s + 9-s − 5.65·10-s − 0.658·11-s − 2.53·12-s + 2.12·14-s + 2.65·15-s − 2.65·16-s + 6.12·17-s + 2.12·18-s + 5.25·19-s − 6.72·20-s − 21-s − 1.40·22-s − 3.53·23-s − 1.12·24-s + 2.06·25-s − 27-s + 2.53·28-s + 0.871·29-s + 5.65·30-s + ⋯ |
L(s) = 1 | + 1.50·2-s − 0.577·3-s + 1.26·4-s − 1.18·5-s − 0.868·6-s + 0.377·7-s + 0.398·8-s + 0.333·9-s − 1.78·10-s − 0.198·11-s − 0.730·12-s + 0.568·14-s + 0.686·15-s − 0.664·16-s + 1.48·17-s + 0.501·18-s + 1.20·19-s − 1.50·20-s − 0.218·21-s − 0.298·22-s − 0.736·23-s − 0.230·24-s + 0.413·25-s − 0.192·27-s + 0.478·28-s + 0.161·29-s + 1.03·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.868205079\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.868205079\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 11 | \( 1 + 0.658T + 11T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 - 0.871T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 3.53T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 4.27T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 9.51T + 89T^{2} \) |
| 97 | \( 1 - 2.97T + 97T^{2} \) |
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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
−8.225709231655387087796056307004, −7.56759527048859250980105278308, −7.06033927847361001351229185354, −5.94219467727618078632015996155, −5.43867242921886512927979216361, −4.80042369375054171090988980058, −3.82827455629773388732128290548, −3.56391432891481177728265619499, −2.36129093888212803844827113574, −0.821651425128952371907217313136,
0.821651425128952371907217313136, 2.36129093888212803844827113574, 3.56391432891481177728265619499, 3.82827455629773388732128290548, 4.80042369375054171090988980058, 5.43867242921886512927979216361, 5.94219467727618078632015996155, 7.06033927847361001351229185354, 7.56759527048859250980105278308, 8.225709231655387087796056307004