| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 0.618·7-s − 8-s + 9-s − 1.61·11-s − 12-s − 4.85·13-s − 0.618·14-s + 16-s − 0.763·17-s − 18-s + 5.85·19-s − 0.618·21-s + 1.61·22-s − 4.85·23-s + 24-s + 4.85·26-s − 27-s + 0.618·28-s − 2.76·29-s − 2.47·31-s − 32-s + 1.61·33-s + 0.763·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.233·7-s − 0.353·8-s + 0.333·9-s − 0.487·11-s − 0.288·12-s − 1.34·13-s − 0.165·14-s + 0.250·16-s − 0.185·17-s − 0.235·18-s + 1.34·19-s − 0.134·21-s + 0.344·22-s − 1.01·23-s + 0.204·24-s + 0.951·26-s − 0.192·27-s + 0.116·28-s − 0.513·29-s − 0.444·31-s − 0.176·32-s + 0.281·33-s + 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 9.56T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 1.61T + 47T^{2} \) |
| 53 | \( 1 + 5.38T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 9.70T + 97T^{2} \) |
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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
−9.852773430018107763351392454132, −9.350765836697224319879775126999, −7.979963908229102351122792734428, −7.53677327273653446071667164609, −6.51298094684943767893618250074, −5.47634101528156222923644704447, −4.64391944384441304322509654789, −3.08363592665244704465937273502, −1.75175652691458209686210142754, 0,
1.75175652691458209686210142754, 3.08363592665244704465937273502, 4.64391944384441304322509654789, 5.47634101528156222923644704447, 6.51298094684943767893618250074, 7.53677327273653446071667164609, 7.979963908229102351122792734428, 9.350765836697224319879775126999, 9.852773430018107763351392454132
