L(s) = 1 | − 3·5-s + 7-s + 3·11-s + 4·13-s + 3·17-s + 4·25-s + 6·29-s − 4·31-s − 3·35-s − 2·37-s − 6·41-s + 43-s − 3·47-s − 6·49-s + 12·53-s − 9·55-s + 6·59-s − 61-s − 12·65-s − 4·67-s − 6·71-s − 7·73-s + 3·77-s + 8·79-s + 12·83-s − 9·85-s + 12·89-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.904·11-s + 1.10·13-s + 0.727·17-s + 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.507·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s − 0.437·47-s − 6/7·49-s + 1.64·53-s − 1.21·55-s + 0.781·59-s − 0.128·61-s − 1.48·65-s − 0.488·67-s − 0.712·71-s − 0.819·73-s + 0.341·77-s + 0.900·79-s + 1.31·83-s − 0.976·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.186586965\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.186586965\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−14.52069919537353, −14.06036870188371, −13.47938618111398, −12.92017371056357, −12.23332794748519, −11.82418337098126, −11.56506835232669, −10.99444170964555, −10.42395361370351, −9.916723032273859, −9.082799743212163, −8.579318020473005, −8.336100555324519, −7.573712692572732, −7.247598908997903, −6.481595159971530, −6.062406338666524, −5.212564481269915, −4.667636823268672, −3.969854336520860, −3.582571885663239, −3.105748941402877, −1.993206817384209, −1.239024518098706, −0.5814224729591374,
0.5814224729591374, 1.239024518098706, 1.993206817384209, 3.105748941402877, 3.582571885663239, 3.969854336520860, 4.667636823268672, 5.212564481269915, 6.062406338666524, 6.481595159971530, 7.247598908997903, 7.573712692572732, 8.336100555324519, 8.579318020473005, 9.082799743212163, 9.916723032273859, 10.42395361370351, 10.99444170964555, 11.56506835232669, 11.82418337098126, 12.23332794748519, 12.92017371056357, 13.47938618111398, 14.06036870188371, 14.52069919537353