| L(s) = 1 | − 3·5-s − 11-s + 3·13-s + 6·17-s − 19-s + 4·23-s + 4·25-s − 5·29-s + 4·31-s + 5·37-s + 2·41-s + 6·43-s − 47-s + 2·53-s + 3·55-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 + 3·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.301·11-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 + 0.821·37-s + 0.312·41-s + 0.914·43-s − 0.145·47-s + 0.274·53-s + 0.404·55-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 + 0.307·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.001004762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001004762\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 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
−14.87956007012909, −14.42167898303108, −13.83597079458313, −13.03823443551894, −12.83946990391971, −12.11301024673547, −11.69620660649105, −11.15608393297049, −10.80754363928316, −10.08587595400035, −9.540687115937825, −8.835968942676390, −8.316053226510724, −7.775265657086100, −7.492801619412210, −6.747516079099561, −6.108625923244390, −5.404352087749931, −4.906132513487482, −3.921228631639840, −3.810060712062528, −3.039088463304321, −2.306281148561951, −1.136771067449227, −0.6161390039464464,
0.6161390039464464, 1.136771067449227, 2.306281148561951, 3.039088463304321, 3.810060712062528, 3.921228631639840, 4.906132513487482, 5.404352087749931, 6.108625923244390, 6.747516079099561, 7.492801619412210, 7.775265657086100, 8.316053226510724, 8.835968942676390, 9.540687115937825, 10.08587595400035, 10.80754363928316, 11.15608393297049, 11.69620660649105, 12.11301024673547, 12.83946990391971, 13.03823443551894, 13.83597079458313, 14.42167898303108, 14.87956007012909
