| L(s) = 1 | − 7-s + 2·11-s − 2·13-s − 4·17-s − 4·19-s + 6·23-s − 5·25-s + 2·29-s − 6·37-s − 8·41-s − 8·43-s + 4·47-s + 49-s + 6·53-s − 14·61-s + 4·67-s + 2·71-s − 2·73-s − 2·77-s + 4·79-s − 12·83-s + 2·91-s + 6·97-s − 12·101-s − 8·103-s − 6·107-s − 18·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.917·19-s + 1.25·23-s − 25-s + 0.371·29-s − 0.986·37-s − 1.24·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.79·61-s + 0.488·67-s + 0.237·71-s − 0.234·73-s − 0.227·77-s + 0.450·79-s − 1.31·83-s + 0.209·91-s + 0.609·97-s − 1.19·101-s − 0.788·103-s − 0.580·107-s − 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.839220827010344299826518365354, −8.080392159110006302593970355953, −6.90671158809997641288656761731, −6.66151764975017501503964240993, −5.54481409757596501307767728969, −4.64683107259554774725500435991, −3.80855384525275899005212589296, −2.76047872572818436273384792924, −1.67384789151853928239355705819, 0,
1.67384789151853928239355705819, 2.76047872572818436273384792924, 3.80855384525275899005212589296, 4.64683107259554774725500435991, 5.54481409757596501307767728969, 6.66151764975017501503964240993, 6.90671158809997641288656761731, 8.080392159110006302593970355953, 8.839220827010344299826518365354
