| L(s) = 1 | − 5-s − 4.89·7-s + 4.89·11-s − 3·13-s + 5·17-s − 4.89·19-s − 4.89·23-s − 4·25-s − 5·29-s + 4.89·35-s + 5·37-s − 2·41-s − 4.89·43-s + 9.79·47-s + 16.9·49-s − 2·53-s − 4.89·55-s − 9.79·59-s + 13·61-s + 3·65-s − 4.89·67-s + 4.89·71-s + 3·73-s − 23.9·77-s + 14.6·79-s − 9.79·83-s − 5·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.85·7-s + 1.47·11-s − 0.832·13-s + 1.21·17-s − 1.12·19-s − 1.02·23-s − 0.800·25-s − 0.928·29-s + 0.828·35-s + 0.821·37-s − 0.312·41-s − 0.747·43-s + 1.42·47-s + 2.42·49-s − 0.274·53-s − 0.660·55-s − 1.27·59-s + 1.66·61-s + 0.372·65-s − 0.598·67-s + 0.581·71-s + 0.351·73-s − 2.73·77-s + 1.65·79-s − 1.07·83-s − 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9516580127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9516580127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.150917327539827677008778213859, −7.42033465512105785760451530098, −6.69268498541615455265485688743, −6.18946316145061602789721299054, −5.50491567628131367817655087555, −4.12718724321880396199702035125, −3.83897759561025808591519441554, −3.00514277623684474839794654782, −1.94672129062779631945298853098, −0.50875853629951289157107948415,
0.50875853629951289157107948415, 1.94672129062779631945298853098, 3.00514277623684474839794654782, 3.83897759561025808591519441554, 4.12718724321880396199702035125, 5.50491567628131367817655087555, 6.18946316145061602789721299054, 6.69268498541615455265485688743, 7.42033465512105785760451530098, 8.150917327539827677008778213859
