| L(s) = 1 | + 1.41·5-s + 4·11-s − 4.24·13-s + 1.41·17-s + 2.82·19-s − 4·23-s − 2.99·25-s − 8·29-s − 8·37-s − 7.07·41-s + 4·43-s − 5.65·47-s − 10·53-s + 5.65·55-s − 14.1·59-s + 7.07·61-s − 6·65-s + 7.07·73-s − 8·79-s + 14.1·83-s + 2.00·85-s + 7.07·89-s + 4.00·95-s + 1.41·97-s − 12.7·101-s − 11.3·103-s + 8·107-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 1.20·11-s − 1.17·13-s + 0.342·17-s + 0.648·19-s − 0.834·23-s − 0.599·25-s − 1.48·29-s − 1.31·37-s − 1.10·41-s + 0.609·43-s − 0.825·47-s − 1.37·53-s + 0.762·55-s − 1.84·59-s + 0.905·61-s − 0.744·65-s + 0.827·73-s − 0.900·79-s + 1.55·83-s + 0.216·85-s + 0.749·89-s + 0.410·95-s + 0.143·97-s − 1.26·101-s − 1.11·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7.07T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−7.55968781945448063860599914089, −6.87722523157641063703877531785, −6.19334204176463672546690709389, −5.48857380834600276585819704535, −4.84591752972666120137644881798, −3.88877022766436247977388013707, −3.23156724846868573201183657511, −2.07588384910867299448030726707, −1.50769343468274766793415940132, 0,
1.50769343468274766793415940132, 2.07588384910867299448030726707, 3.23156724846868573201183657511, 3.88877022766436247977388013707, 4.84591752972666120137644881798, 5.48857380834600276585819704535, 6.19334204176463672546690709389, 6.87722523157641063703877531785, 7.55968781945448063860599914089
