L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 3·7-s + 9-s + 11-s − 2·12-s + 2·13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s + 6·19-s − 3·21-s − 2·22-s − 3·23-s − 4·26-s − 27-s + 6·28-s + 3·29-s + 2·31-s + 8·32-s − 33-s + 4·34-s + 2·36-s + 7·37-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s + 1.37·19-s − 0.654·21-s − 0.426·22-s − 0.625·23-s − 0.784·26-s − 0.192·27-s + 1.13·28-s + 0.557·29-s + 0.359·31-s + 1.41·32-s − 0.174·33-s + 0.685·34-s + 1/3·36-s + 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9425185637\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9425185637\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.602584629951561113830733197974, −7.78091108170505847598252560033, −7.51664253316875775477338396262, −6.47471788578892689826689313513, −5.74730307755447516496194871617, −4.73279478889276445796998568082, −4.11797623123414630335873886332, −2.60023057208825044992396591035, −1.49548225835488390565202219199, −0.818394894647382605599021148107,
0.818394894647382605599021148107, 1.49548225835488390565202219199, 2.60023057208825044992396591035, 4.11797623123414630335873886332, 4.73279478889276445796998568082, 5.74730307755447516496194871617, 6.47471788578892689826689313513, 7.51664253316875775477338396262, 7.78091108170505847598252560033, 8.602584629951561113830733197974