| L(s) = 1 | − 1.48·2-s + 2.16·3-s + 0.193·4-s + 2.23·5-s − 3.20·6-s − 3.28·7-s + 2.67·8-s + 1.68·9-s − 3.30·10-s + 2.94·11-s + 0.418·12-s + 5.66·13-s + 4.86·14-s + 4.83·15-s − 4.34·16-s − 0.266·17-s − 2.49·18-s + 3.40·19-s + 0.432·20-s − 7.10·21-s − 4.36·22-s − 1.09·23-s + 5.79·24-s − 0.0147·25-s − 8.39·26-s − 2.84·27-s − 0.635·28-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 1.24·3-s + 0.0967·4-s + 0.998·5-s − 1.30·6-s − 1.24·7-s + 0.945·8-s + 0.561·9-s − 1.04·10-s + 0.887·11-s + 0.120·12-s + 1.57·13-s + 1.29·14-s + 1.24·15-s − 1.08·16-s − 0.0645·17-s − 0.588·18-s + 0.781·19-s + 0.0966·20-s − 1.55·21-s − 0.929·22-s − 0.228·23-s + 1.18·24-s − 0.00294·25-s − 1.64·26-s − 0.547·27-s − 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.625449793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625449793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 13 | \( 1 - 5.66T + 13T^{2} \) |
| 17 | \( 1 + 0.266T + 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 + 1.44T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 + 6.21T + 67T^{2} \) |
| 71 | \( 1 + 0.971T + 71T^{2} \) |
| 73 | \( 1 - 0.870T + 73T^{2} \) |
| 79 | \( 1 + 1.18T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 - 3.62T + 89T^{2} \) |
| 97 | \( 1 - 7.43T + 97T^{2} \) |
| 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
−9.492595966544262645224097455748, −8.967726309298723236944495067172, −8.321288892494820732249471019276, −7.42047074539510686495778028026, −6.43440602243154843064918870126, −5.75344917094015121773396949248, −4.08402044478253206362336385033, −3.35898684893703827112196958518, −2.19689510638038358328228720238, −1.10393713392436699541066547774,
1.10393713392436699541066547774, 2.19689510638038358328228720238, 3.35898684893703827112196958518, 4.08402044478253206362336385033, 5.75344917094015121773396949248, 6.43440602243154843064918870126, 7.42047074539510686495778028026, 8.321288892494820732249471019276, 8.967726309298723236944495067172, 9.492595966544262645224097455748
