| L(s) = 1 | + 3-s + 1.41·5-s + 1.41·7-s + 9-s + 2·11-s + 1.41·15-s + 2·17-s + 4·19-s + 1.41·21-s + 2.82·23-s − 2.99·25-s + 27-s − 9.89·29-s − 7.07·31-s + 2·33-s + 2.00·35-s + 8.48·37-s + 6·41-s + 8·43-s + 1.41·45-s + 2.82·47-s − 5·49-s + 2·51-s + 1.41·53-s + 2.82·55-s + 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.632·5-s + 0.534·7-s + 0.333·9-s + 0.603·11-s + 0.365·15-s + 0.485·17-s + 0.917·19-s + 0.308·21-s + 0.589·23-s − 0.599·25-s + 0.192·27-s − 1.83·29-s − 1.27·31-s + 0.348·33-s + 0.338·35-s + 1.39·37-s + 0.937·41-s + 1.21·43-s + 0.210·45-s + 0.412·47-s − 0.714·49-s + 0.280·51-s + 0.194·53-s + 0.381·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.669346980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669346980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.344362040845107118359055715455, −8.898230127423678495459227882582, −7.63614701002364541388417247504, −7.39560216304176002028261402091, −6.04534803929674304450213973961, −5.45215002417403430665749076969, −4.30824110962678800119848532316, −3.41643432170277313086471770107, −2.25487913579769268532888068223, −1.27889856704408485626853232664,
1.27889856704408485626853232664, 2.25487913579769268532888068223, 3.41643432170277313086471770107, 4.30824110962678800119848532316, 5.45215002417403430665749076969, 6.04534803929674304450213973961, 7.39560216304176002028261402091, 7.63614701002364541388417247504, 8.898230127423678495459227882582, 9.344362040845107118359055715455
