| L(s) = 1 | + 2.72·2-s + 2.09·3-s + 5.44·4-s − 3.87·5-s + 5.71·6-s + 0.707·7-s + 9.39·8-s + 1.38·9-s − 10.5·10-s + 11-s + 11.4·12-s − 5.09·13-s + 1.93·14-s − 8.12·15-s + 14.7·16-s + 5.13·17-s + 3.78·18-s − 1.83·19-s − 21.1·20-s + 1.48·21-s + 2.72·22-s − 5.52·23-s + 19.6·24-s + 10.0·25-s − 13.8·26-s − 3.38·27-s + 3.85·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 1.20·3-s + 2.72·4-s − 1.73·5-s + 2.33·6-s + 0.267·7-s + 3.32·8-s + 0.461·9-s − 3.34·10-s + 0.301·11-s + 3.29·12-s − 1.41·13-s + 0.515·14-s − 2.09·15-s + 3.68·16-s + 1.24·17-s + 0.890·18-s − 0.419·19-s − 4.72·20-s + 0.323·21-s + 0.581·22-s − 1.15·23-s + 4.01·24-s + 2.00·25-s − 2.72·26-s − 0.650·27-s + 0.727·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.262344181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.262344181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 - 0.707T + 7T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 - 5.13T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 + 6.78T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 - 5.48T + 59T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 8.09T + 71T^{2} \) |
| 73 | \( 1 + 0.965T + 73T^{2} \) |
| 79 | \( 1 - 7.96T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−10.98522983522042101131117917896, −9.771979683154317419740991245890, −8.284394835611168469595862895442, −7.60494987951960740855610891265, −7.19260582179375620228652476472, −5.72629298541482891143082785358, −4.60595691571991808236802807603, −3.87084996489562370305149657176, −3.24210195051668055667694557508, −2.18740406900274472852131745159,
2.18740406900274472852131745159, 3.24210195051668055667694557508, 3.87084996489562370305149657176, 4.60595691571991808236802807603, 5.72629298541482891143082785358, 7.19260582179375620228652476472, 7.60494987951960740855610891265, 8.284394835611168469595862895442, 9.771979683154317419740991245890, 10.98522983522042101131117917896
