| L(s) = 1 | − 7-s − 6·11-s + 4·13-s + 6·17-s + 2·19-s − 6·29-s − 10·31-s − 2·37-s + 6·41-s + 4·43-s + 49-s − 12·53-s + 14·61-s + 4·67-s − 6·71-s + 4·73-s + 6·77-s − 16·79-s − 12·83-s − 6·89-s − 4·91-s + 16·97-s − 6·101-s + 16·103-s + 2·109-s − 6·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.80·11-s + 1.10·13-s + 1.45·17-s + 0.458·19-s − 1.11·29-s − 1.79·31-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 1.64·53-s + 1.79·61-s + 0.488·67-s − 0.712·71-s + 0.468·73-s + 0.683·77-s − 1.80·79-s − 1.31·83-s − 0.635·89-s − 0.419·91-s + 1.62·97-s − 0.597·101-s + 1.57·103-s + 0.191·109-s − 0.550·119-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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.56392139303330492554341277612, −7.28050951123702380694700794552, −6.00653890840457663914926140231, −5.63206027677827909273739177339, −5.00586811678844163420390121787, −3.79922513033190762033835494220, −3.28838352842111983183465755756, −2.39173644572891473203638331037, −1.27905609811051026287198901089, 0,
1.27905609811051026287198901089, 2.39173644572891473203638331037, 3.28838352842111983183465755756, 3.79922513033190762033835494220, 5.00586811678844163420390121787, 5.63206027677827909273739177339, 6.00653890840457663914926140231, 7.28050951123702380694700794552, 7.56392139303330492554341277612
