L(s) = 1 | + 1.36·2-s + 3.15·3-s − 6.14·4-s + 3.03·5-s + 4.29·6-s − 7.94·7-s − 19.2·8-s − 17.0·9-s + 4.12·10-s + 27.6·11-s − 19.3·12-s + 58.1·13-s − 10.8·14-s + 9.56·15-s + 22.9·16-s − 17·17-s − 23.2·18-s + 89.1·19-s − 18.6·20-s − 25.0·21-s + 37.5·22-s − 115.·23-s − 60.7·24-s − 115.·25-s + 79.1·26-s − 138.·27-s + 48.8·28-s + ⋯ |
L(s) = 1 | + 0.481·2-s + 0.607·3-s − 0.768·4-s + 0.271·5-s + 0.292·6-s − 0.428·7-s − 0.851·8-s − 0.631·9-s + 0.130·10-s + 0.756·11-s − 0.466·12-s + 1.23·13-s − 0.206·14-s + 0.164·15-s + 0.358·16-s − 0.242·17-s − 0.303·18-s + 1.07·19-s − 0.208·20-s − 0.260·21-s + 0.364·22-s − 1.04·23-s − 0.516·24-s − 0.926·25-s + 0.596·26-s − 0.990·27-s + 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.259736665\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.259736665\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + 17T \) |
good | 2 | \( 1 - 1.36T + 8T^{2} \) |
| 3 | \( 1 - 3.15T + 27T^{2} \) |
| 5 | \( 1 - 3.03T + 125T^{2} \) |
| 7 | \( 1 + 7.94T + 343T^{2} \) |
| 11 | \( 1 - 27.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 89.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 527.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 53.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 52.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 295.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 720.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 813.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 794.T + 9.12e5T^{2} \) |
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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
−18.47951223268626911027103487299, −17.26837424769575811362802477693, −15.56194346425825622226188432957, −14.05477192401478735713806224842, −13.53903149732419800522825997454, −11.82587820016891111095247821777, −9.632473151912304259544737115312, −8.468797931306279695202548895799, −5.93382345550704463877322310005, −3.63824331318068623675418956800,
3.63824331318068623675418956800, 5.93382345550704463877322310005, 8.468797931306279695202548895799, 9.632473151912304259544737115312, 11.82587820016891111095247821777, 13.53903149732419800522825997454, 14.05477192401478735713806224842, 15.56194346425825622226188432957, 17.26837424769575811362802477693, 18.47951223268626911027103487299