| L(s) = 1 | + 5-s + 3.82·11-s + 3.58·13-s − 6.41·17-s + 3.65·19-s − 0.585·23-s + 25-s − 6.65·29-s + 4.58·31-s − 3.41·37-s − 0.585·41-s + 11.6·43-s + 8.89·47-s + 3.75·53-s + 3.82·55-s − 3.41·59-s + 5.17·61-s + 3.58·65-s − 11.0·67-s − 6.48·71-s + 5.17·73-s + 13.1·79-s + 8·83-s − 6.41·85-s − 16.9·89-s + 3.65·95-s + 15.7·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.15·11-s + 0.994·13-s − 1.55·17-s + 0.838·19-s − 0.122·23-s + 0.200·25-s − 1.23·29-s + 0.823·31-s − 0.561·37-s − 0.0914·41-s + 1.77·43-s + 1.29·47-s + 0.516·53-s + 0.516·55-s − 0.444·59-s + 0.662·61-s + 0.444·65-s − 1.35·67-s − 0.769·71-s + 0.605·73-s + 1.47·79-s + 0.878·83-s − 0.695·85-s − 1.79·89-s + 0.375·95-s + 1.59·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599737734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599737734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 + 0.585T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−7.62669751411970789083017607910, −7.05132693061622322299786661402, −6.23983838494549288897367619282, −5.93348618218933343568554059532, −4.96276053846876674309562392993, −4.13266352149547246388917167820, −3.61871746415233999617609018727, −2.54737025569350096294057922665, −1.72213895795803709345824939178, −0.816994259288985305457595229570,
0.816994259288985305457595229570, 1.72213895795803709345824939178, 2.54737025569350096294057922665, 3.61871746415233999617609018727, 4.13266352149547246388917167820, 4.96276053846876674309562392993, 5.93348618218933343568554059532, 6.23983838494549288897367619282, 7.05132693061622322299786661402, 7.62669751411970789083017607910
