L(s) = 1 | − 3-s + 9-s + 11-s + 13-s + 5·17-s − 4·23-s − 5·25-s − 27-s − 6·31-s − 33-s + 2·37-s − 39-s − 8·41-s − 2·43-s − 10·47-s − 7·49-s − 5·51-s + 53-s − 3·59-s − 4·61-s + 8·67-s + 4·69-s + 15·71-s + 14·73-s + 5·75-s − 79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 1.21·17-s − 0.834·23-s − 25-s − 0.192·27-s − 1.07·31-s − 0.174·33-s + 0.328·37-s − 0.160·39-s − 1.24·41-s − 0.304·43-s − 1.45·47-s − 49-s − 0.700·51-s + 0.137·53-s − 0.390·59-s − 0.512·61-s + 0.977·67-s + 0.481·69-s + 1.78·71-s + 1.63·73-s + 0.577·75-s − 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.95995443151206, −13.61604983985804, −13.02926914725529, −12.43772054927030, −12.11855074667790, −11.62085360757283, −11.13985386395374, −10.67697851253746, −10.05832591588173, −9.554742154521652, −9.388362489265939, −8.340776737798114, −8.048252134675852, −7.612439575371849, −6.787180457429400, −6.461859483567891, −5.891967456155195, −5.270279159809410, −4.981478709167453, −4.082182732019950, −3.607850288550559, −3.179171078219326, −2.048261724902477, −1.690109807513627, −0.8217348079906437, 0,
0.8217348079906437, 1.690109807513627, 2.048261724902477, 3.179171078219326, 3.607850288550559, 4.082182732019950, 4.981478709167453, 5.270279159809410, 5.891967456155195, 6.461859483567891, 6.787180457429400, 7.612439575371849, 8.048252134675852, 8.340776737798114, 9.388362489265939, 9.554742154521652, 10.05832591588173, 10.67697851253746, 11.13985386395374, 11.62085360757283, 12.11855074667790, 12.43772054927030, 13.02926914725529, 13.61604983985804, 13.95995443151206