L(s) = 1 | − 3-s + 9-s − 2·11-s + 13-s + 19-s + 2·23-s − 27-s − 8·29-s + 2·33-s + 7·37-s − 39-s − 2·41-s + 4·43-s − 4·53-s − 57-s − 5·61-s − 67-s − 2·69-s − 7·73-s + 79-s + 81-s + 8·83-s + 8·87-s + 12·89-s − 3·97-s − 2·99-s + 101-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.229·19-s + 0.417·23-s − 0.192·27-s − 1.48·29-s + 0.348·33-s + 1.15·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.549·53-s − 0.132·57-s − 0.640·61-s − 0.122·67-s − 0.240·69-s − 0.819·73-s + 0.112·79-s + 1/9·81-s + 0.878·83-s + 0.857·87-s + 1.27·89-s − 0.304·97-s − 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−15.52844137263903, −14.86499935266791, −14.52795565530585, −13.67150496156607, −13.23909849775636, −12.84051303707669, −12.22286007395442, −11.63505188260479, −11.10187686662108, −10.73120747199022, −10.10659472234253, −9.470898848600228, −9.032370569529342, −8.276593616320197, −7.551860532784977, −7.344537739604574, −6.371066348006466, −6.009948644979948, −5.304549302083648, −4.820188007353123, −4.072361558539929, −3.401760695396933, −2.622692687985859, −1.815923926121236, −0.9487973607140939, 0,
0.9487973607140939, 1.815923926121236, 2.622692687985859, 3.401760695396933, 4.072361558539929, 4.820188007353123, 5.304549302083648, 6.009948644979948, 6.371066348006466, 7.344537739604574, 7.551860532784977, 8.276593616320197, 9.032370569529342, 9.470898848600228, 10.10659472234253, 10.73120747199022, 11.10187686662108, 11.63505188260479, 12.22286007395442, 12.84051303707669, 13.23909849775636, 13.67150496156607, 14.52795565530585, 14.86499935266791, 15.52844137263903