| L(s) = 1 | + 7-s − 4·11-s + 2·13-s − 2·17-s − 2·19-s + 6·23-s − 6·29-s + 6·31-s − 4·37-s − 4·43-s − 4·47-s + 49-s + 2·53-s − 4·59-s − 2·61-s − 12·67-s + 8·71-s + 14·73-s − 4·77-s + 16·79-s − 16·83-s − 16·89-s + 2·91-s − 14·97-s − 8·101-s − 18·107-s + 18·109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.458·19-s + 1.25·23-s − 1.11·29-s + 1.07·31-s − 0.657·37-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.274·53-s − 0.520·59-s − 0.256·61-s − 1.46·67-s + 0.949·71-s + 1.63·73-s − 0.455·77-s + 1.80·79-s − 1.75·83-s − 1.69·89-s + 0.209·91-s − 1.42·97-s − 0.796·101-s − 1.74·107-s + 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.77159659258027290017412040586, −6.98613121146011876526242358338, −6.32575758670515486132565847687, −5.40176126604474940190397954926, −4.92762065880481494145037962972, −4.06397909981608459085969054295, −3.12889109626994772042457064021, −2.35227648687309806203828311827, −1.34463390596287361643537502963, 0,
1.34463390596287361643537502963, 2.35227648687309806203828311827, 3.12889109626994772042457064021, 4.06397909981608459085969054295, 4.92762065880481494145037962972, 5.40176126604474940190397954926, 6.32575758670515486132565847687, 6.98613121146011876526242358338, 7.77159659258027290017412040586
