L(s) = 1 | + 3·5-s + 7-s + 3·13-s + 6·17-s − 19-s + 4·23-s + 4·25-s + 5·29-s − 4·31-s + 3·35-s + 5·37-s + 2·41-s − 6·43-s + 47-s + 49-s + 2·53-s − 7·59-s + 14·61-s + 9·65-s + 67-s − 6·71-s + 17·73-s − 2·79-s + 14·83-s + 18·85-s + 6·89-s + 3·91-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.377·7-s + 0.832·13-s + 1.45·17-s − 0.229·19-s + 0.834·23-s + 4/5·25-s + 0.928·29-s − 0.718·31-s + 0.507·35-s + 0.821·37-s + 0.312·41-s − 0.914·43-s + 0.145·47-s + 1/7·49-s + 0.274·53-s − 0.911·59-s + 1.79·61-s + 1.11·65-s + 0.122·67-s − 0.712·71-s + 1.98·73-s − 0.225·79-s + 1.53·83-s + 1.95·85-s + 0.635·89-s + 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.788748780\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.788748780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.27986832598894, −13.71852985711912, −13.43924414213734, −12.78673658869915, −12.43999874230310, −11.71004416880997, −11.21122091776004, −10.65534196891264, −10.16136442752130, −9.779453386713416, −9.124384835977450, −8.780432518326984, −8.012139945528870, −7.675801452129525, −6.768651774215801, −6.399178922022032, −5.801431740313109, −5.288619762209317, −4.909334842448314, −3.981990832977267, −3.391732751275484, −2.687312950784510, −2.036319168224919, −1.320185616112825, −0.8193754050923434,
0.8193754050923434, 1.320185616112825, 2.036319168224919, 2.687312950784510, 3.391732751275484, 3.981990832977267, 4.909334842448314, 5.288619762209317, 5.801431740313109, 6.399178922022032, 6.768651774215801, 7.675801452129525, 8.012139945528870, 8.780432518326984, 9.124384835977450, 9.779453386713416, 10.16136442752130, 10.65534196891264, 11.21122091776004, 11.71004416880997, 12.43999874230310, 12.78673658869915, 13.43924414213734, 13.71852985711912, 14.27986832598894