| L(s) = 1 | + 1.41·5-s + 2·7-s − 1.41·11-s − 1.41·13-s − 2·17-s − 4.24·19-s − 6·23-s − 2.99·25-s + 4.24·29-s + 8·31-s + 2.82·35-s + 4.24·37-s + 7.07·43-s − 8·47-s − 3·49-s − 7.07·53-s − 2.00·55-s − 4.24·59-s + 12.7·61-s − 2.00·65-s − 7.07·67-s − 10·71-s + 4·73-s − 2.82·77-s − 1.41·83-s − 2.82·85-s − 4·89-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 0.755·7-s − 0.426·11-s − 0.392·13-s − 0.485·17-s − 0.973·19-s − 1.25·23-s − 0.599·25-s + 0.787·29-s + 1.43·31-s + 0.478·35-s + 0.697·37-s + 1.07·43-s − 1.16·47-s − 0.428·49-s − 0.971·53-s − 0.269·55-s − 0.552·59-s + 1.62·61-s − 0.248·65-s − 0.863·67-s − 1.18·71-s + 0.468·73-s − 0.322·77-s − 0.155·83-s − 0.306·85-s − 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.51492538936526841986057630477, −6.45742913161854301633481661620, −6.16449417391641614736057879116, −5.26907252264224647150774383578, −4.58477741635161640209811755153, −4.05969940102495599980727480232, −2.78097724796515450927844404860, −2.21747932060423342208001006657, −1.38943510269200836088248952408, 0,
1.38943510269200836088248952408, 2.21747932060423342208001006657, 2.78097724796515450927844404860, 4.05969940102495599980727480232, 4.58477741635161640209811755153, 5.26907252264224647150774383578, 6.16449417391641614736057879116, 6.45742913161854301633481661620, 7.51492538936526841986057630477
