| L(s) = 1 | − 1.79·3-s − 5-s − 2.79·7-s + 0.208·9-s − 3.79·11-s + 1.20·13-s + 1.79·15-s − 3.79·17-s − 1.20·19-s + 5·21-s − 23-s + 25-s + 5.00·27-s − 1.58·29-s − 10.3·31-s + 6.79·33-s + 2.79·35-s − 4·37-s − 2.16·39-s − 2.20·41-s + 7.16·43-s − 0.208·45-s + 13.5·47-s + 0.791·49-s + 6.79·51-s + 6·53-s + 3.79·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s − 0.447·5-s − 1.05·7-s + 0.0695·9-s − 1.14·11-s + 0.335·13-s + 0.462·15-s − 0.919·17-s − 0.277·19-s + 1.09·21-s − 0.208·23-s + 0.200·25-s + 0.962·27-s − 0.293·29-s − 1.86·31-s + 1.18·33-s + 0.471·35-s − 0.657·37-s − 0.346·39-s − 0.344·41-s + 1.09·43-s − 0.0311·45-s + 1.98·47-s + 0.113·49-s + 0.950·51-s + 0.824·53-s + 0.511·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3873942648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3873942648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 29 | \( 1 + 1.58T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 - 7.16T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 3.16T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.192558564449213104883704394554, −8.572613416526234713200387429236, −7.45851576494995521440588353022, −6.84408143774689620859325903109, −5.91050763643489754797739481778, −5.43262524105306635957983829240, −4.35724510353962739433974942985, −3.40263296772532377232184229017, −2.31289659133432238693076633442, −0.41599783352405562787186114485,
0.41599783352405562787186114485, 2.31289659133432238693076633442, 3.40263296772532377232184229017, 4.35724510353962739433974942985, 5.43262524105306635957983829240, 5.91050763643489754797739481778, 6.84408143774689620859325903109, 7.45851576494995521440588353022, 8.572613416526234713200387429236, 9.192558564449213104883704394554
