| L(s) = 1 | + 3.41·5-s + 0.828·7-s + 2.82·11-s + 0.828·13-s + 0.585·17-s − 5.65·19-s + 2.58·23-s + 6.65·25-s + 10.2·29-s − 1.17·31-s + 2.82·35-s − 37-s + 0.343·41-s + 2.82·43-s + 5.65·47-s − 6.31·49-s + 7.17·53-s + 9.65·55-s − 4.24·59-s − 9.31·61-s + 2.82·65-s − 0.828·67-s − 8·71-s + 8·73-s + 2.34·77-s + 2.34·79-s − 16.4·83-s + ⋯ |
| L(s) = 1 | + 1.52·5-s + 0.313·7-s + 0.852·11-s + 0.229·13-s + 0.142·17-s − 1.29·19-s + 0.539·23-s + 1.33·25-s + 1.90·29-s − 0.210·31-s + 0.478·35-s − 0.164·37-s + 0.0535·41-s + 0.431·43-s + 0.825·47-s − 0.901·49-s + 0.985·53-s + 1.30·55-s − 0.552·59-s − 1.19·61-s + 0.350·65-s − 0.101·67-s − 0.949·71-s + 0.936·73-s + 0.267·77-s + 0.263·79-s − 1.80·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.786710808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.786710808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 0.585T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 0.828T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.906871857739975549094719065495, −8.335147139073202048586367610641, −7.14874486987630258692906760413, −6.37860711243168680742285091339, −5.95402724350124468029955057013, −4.96904902888304444661653108831, −4.21220662622938707632058833731, −2.95372989549018036523062830306, −2.03065987467896487125104360531, −1.15158556770127374021296500838,
1.15158556770127374021296500838, 2.03065987467896487125104360531, 2.95372989549018036523062830306, 4.21220662622938707632058833731, 4.96904902888304444661653108831, 5.95402724350124468029955057013, 6.37860711243168680742285091339, 7.14874486987630258692906760413, 8.335147139073202048586367610641, 8.906871857739975549094719065495
