L(s) = 1 | + 2-s + 3.30·3-s + 4-s − 2.30·5-s + 3.30·6-s + 2.60·7-s + 8-s + 7.90·9-s − 2.30·10-s + 3.30·12-s − 1.30·13-s + 2.60·14-s − 7.60·15-s + 16-s + 6·17-s + 7.90·18-s − 2·19-s − 2.30·20-s + 8.60·21-s + 3.90·23-s + 3.30·24-s + 0.302·25-s − 1.30·26-s + 16.2·27-s + 2.60·28-s + 3.90·29-s − 7.60·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.90·3-s + 0.5·4-s − 1.02·5-s + 1.34·6-s + 0.984·7-s + 0.353·8-s + 2.63·9-s − 0.728·10-s + 0.953·12-s − 0.361·13-s + 0.696·14-s − 1.96·15-s + 0.250·16-s + 1.45·17-s + 1.86·18-s − 0.458·19-s − 0.514·20-s + 1.87·21-s + 0.814·23-s + 0.674·24-s + 0.0605·25-s − 0.255·26-s + 3.11·27-s + 0.492·28-s + 0.725·29-s − 1.38·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8954 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8954 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.274716478\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.274716478\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−7.85127799871097651693780737745, −7.29225082054417778499650203958, −6.68551520258095914812797496216, −5.37334396147741940951367562840, −4.67009263454832620181363519777, −4.10654327673380068897849834263, −3.40621621974509071652066146448, −2.90931643311250936308460054015, −1.99135083638354144818915967395, −1.17718697634270251882419943489,
1.17718697634270251882419943489, 1.99135083638354144818915967395, 2.90931643311250936308460054015, 3.40621621974509071652066146448, 4.10654327673380068897849834263, 4.67009263454832620181363519777, 5.37334396147741940951367562840, 6.68551520258095914812797496216, 7.29225082054417778499650203958, 7.85127799871097651693780737745