| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s + 16-s − 6·17-s + 18-s − 2·24-s + 4·27-s + 6·29-s + 6·31-s + 32-s − 6·34-s + 36-s + 6·37-s − 10·43-s − 12·47-s − 2·48-s − 7·49-s + 12·51-s + 4·54-s + 6·58-s + 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.408·24-s + 0.769·27-s + 1.11·29-s + 1.07·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s − 1.52·43-s − 1.75·47-s − 0.288·48-s − 49-s + 1.68·51-s + 0.544·54-s + 0.787·58-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−6.94349230940694833310944619250, −6.64659726502570707675719407401, −6.10005890269898004905858264821, −5.26876564764173839291367167532, −4.75020181596378963433129210770, −4.17952128706732157660846479119, −3.10613091772616344768231870845, −2.33315797372544332415861764056, −1.18632587390546547137118153299, 0,
1.18632587390546547137118153299, 2.33315797372544332415861764056, 3.10613091772616344768231870845, 4.17952128706732157660846479119, 4.75020181596378963433129210770, 5.26876564764173839291367167532, 6.10005890269898004905858264821, 6.64659726502570707675719407401, 6.94349230940694833310944619250
