| L(s) = 1 | − 5.56·2-s + 3·3-s + 22.9·4-s − 9.37·5-s − 16.6·6-s + 18.3·7-s − 83.3·8-s + 9·9-s + 52.1·10-s + 63.5·11-s + 68.9·12-s − 10.7·13-s − 102.·14-s − 28.1·15-s + 280.·16-s − 79.2·17-s − 50.0·18-s − 15.5·19-s − 215.·20-s + 55.0·21-s − 353.·22-s + 13.8·23-s − 250.·24-s − 37.0·25-s + 59.8·26-s + 27·27-s + 421.·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 0.577·3-s + 2.87·4-s − 0.838·5-s − 1.13·6-s + 0.990·7-s − 3.68·8-s + 0.333·9-s + 1.65·10-s + 1.74·11-s + 1.65·12-s − 0.229·13-s − 1.94·14-s − 0.484·15-s + 4.37·16-s − 1.13·17-s − 0.655·18-s − 0.187·19-s − 2.40·20-s + 0.571·21-s − 3.42·22-s + 0.125·23-s − 2.12·24-s − 0.296·25-s + 0.451·26-s + 0.192·27-s + 2.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9019229200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9019229200\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 + 5.56T + 8T^{2} \) |
| 5 | \( 1 + 9.37T + 125T^{2} \) |
| 7 | \( 1 - 18.3T + 343T^{2} \) |
| 11 | \( 1 - 63.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 210.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 403.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 175.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 73.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 588.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 398.T + 2.26e5T^{2} \) |
| 71 | \( 1 + 2.78T + 3.57e5T^{2} \) |
| 73 | \( 1 - 459.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 956.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 129.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 2.29T + 9.12e5T^{2} \) |
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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
−11.43492410592821246520322649195, −11.08085422562335462819339658691, −9.674017629981333780378476511840, −8.906112799012941877171985309771, −8.196868877424005626828845638064, −7.35839852698617396178073220798, −6.41963039542593252614910073436, −4.05515029901979969572474792011, −2.29819590558449945548713639651, −0.982690703632391886625842958100,
0.982690703632391886625842958100, 2.29819590558449945548713639651, 4.05515029901979969572474792011, 6.41963039542593252614910073436, 7.35839852698617396178073220798, 8.196868877424005626828845638064, 8.906112799012941877171985309771, 9.674017629981333780378476511840, 11.08085422562335462819339658691, 11.43492410592821246520322649195
