L(s) = 1 | + 2-s − 3-s + 4-s − 3.70·5-s − 6-s + 7-s + 8-s + 9-s − 3.70·10-s + 5.70·11-s − 12-s + 13-s + 14-s + 3.70·15-s + 16-s + 3.70·17-s + 18-s + 5.70·19-s − 3.70·20-s − 21-s + 5.70·22-s − 1.70·23-s − 24-s + 8.70·25-s + 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.65·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.17·10-s + 1.71·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.955·15-s + 0.250·16-s + 0.897·17-s + 0.235·18-s + 1.30·19-s − 0.827·20-s − 0.218·21-s + 1.21·22-s − 0.354·23-s − 0.204·24-s + 1.74·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.649556773\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.649556773\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 - 5.70T + 11T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 + 1.70T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 7.70T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 + 3.40T + 79T^{2} \) |
| 83 | \( 1 + 0.596T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−11.43483643089544292602743656076, −10.15243844260323863445322529913, −8.953394879882585085534439410086, −7.81391489647087793893531511872, −7.19826094541585290985540074696, −6.17195708811421168737571013421, −5.03677385107277837904297481321, −4.02564892686347259075982140467, −3.44521090361969176928239126777, −1.18483450496188729564153348359,
1.18483450496188729564153348359, 3.44521090361969176928239126777, 4.02564892686347259075982140467, 5.03677385107277837904297481321, 6.17195708811421168737571013421, 7.19826094541585290985540074696, 7.81391489647087793893531511872, 8.953394879882585085534439410086, 10.15243844260323863445322529913, 11.43483643089544292602743656076