| L(s) = 1 | − 1.21·2-s − 3-s − 0.516·4-s − 1.00·5-s + 1.21·6-s − 0.856·7-s + 3.06·8-s + 9-s + 1.22·10-s − 11-s + 0.516·12-s − 6.03·13-s + 1.04·14-s + 1.00·15-s − 2.70·16-s + 1.06·17-s − 1.21·18-s − 4.50·19-s + 0.519·20-s + 0.856·21-s + 1.21·22-s + 2.56·23-s − 3.06·24-s − 3.98·25-s + 7.35·26-s − 27-s + 0.442·28-s + ⋯ |
| L(s) = 1 | − 0.861·2-s − 0.577·3-s − 0.258·4-s − 0.450·5-s + 0.497·6-s − 0.323·7-s + 1.08·8-s + 0.333·9-s + 0.387·10-s − 0.301·11-s + 0.149·12-s − 1.67·13-s + 0.278·14-s + 0.259·15-s − 0.675·16-s + 0.258·17-s − 0.287·18-s − 1.03·19-s + 0.116·20-s + 0.186·21-s + 0.259·22-s + 0.534·23-s − 0.625·24-s − 0.797·25-s + 1.44·26-s − 0.192·27-s + 0.0835·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2378217853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2378217853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 + 0.856T + 7T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 + 0.470T + 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 - 7.89T + 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 7.48T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 2.39T + 89T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.721694323662760421667510002911, −7.76529877192250878718936982475, −7.50422862560125986660061213331, −6.57928218079541262823123031919, −5.62386093936095021900398196134, −4.70154165335685296444410380723, −4.25010585677572706889924642304, −2.93725780099688680838568871571, −1.76151905635814538277867567265, −0.33854781948092809296409279539,
0.33854781948092809296409279539, 1.76151905635814538277867567265, 2.93725780099688680838568871571, 4.25010585677572706889924642304, 4.70154165335685296444410380723, 5.62386093936095021900398196134, 6.57928218079541262823123031919, 7.50422862560125986660061213331, 7.76529877192250878718936982475, 8.721694323662760421667510002911
