L(s) = 1 | − 2-s − 4-s + 5-s − 7-s + 3·8-s − 10-s − 2·11-s − 5·13-s + 14-s − 16-s + 3·17-s − 2·19-s − 20-s + 2·22-s + 6·23-s − 4·25-s + 5·26-s + 28-s − 5·29-s − 6·31-s − 5·32-s − 3·34-s − 35-s − 3·37-s + 2·38-s + 3·40-s − 10·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.316·10-s − 0.603·11-s − 1.38·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s − 0.223·20-s + 0.426·22-s + 1.25·23-s − 4/5·25-s + 0.980·26-s + 0.188·28-s − 0.928·29-s − 1.07·31-s − 0.883·32-s − 0.514·34-s − 0.169·35-s − 0.493·37-s + 0.324·38-s + 0.474·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.01478718768857914205383739726, −9.580535163753240714686698310753, −8.679478896658523587392022167994, −7.69836590322364312036128660126, −6.98149950262340720530971399684, −5.52643116454046103792293973791, −4.82942330530482617139033746396, −3.38522338168038707417366380666, −1.89538931356427475498208080000, 0,
1.89538931356427475498208080000, 3.38522338168038707417366380666, 4.82942330530482617139033746396, 5.52643116454046103792293973791, 6.98149950262340720530971399684, 7.69836590322364312036128660126, 8.679478896658523587392022167994, 9.580535163753240714686698310753, 10.01478718768857914205383739726