L(s) = 1 | − 3·9-s + 2·13-s − 17-s + 4·19-s − 8·23-s + 2·29-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s − 7·49-s − 6·53-s + 4·59-s − 6·61-s + 4·67-s + 8·71-s − 2·73-s + 9·81-s + 4·83-s − 6·89-s − 18·97-s + 14·101-s − 16·103-s + 16·107-s − 14·109-s − 10·113-s − 6·117-s + ⋯ |
L(s) = 1 | − 9-s + 0.554·13-s − 0.242·17-s + 0.917·19-s − 1.66·23-s + 0.371·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 49-s − 0.824·53-s + 0.520·59-s − 0.768·61-s + 0.488·67-s + 0.949·71-s − 0.234·73-s + 81-s + 0.439·83-s − 0.635·89-s − 1.82·97-s + 1.39·101-s − 1.57·103-s + 1.54·107-s − 1.34·109-s − 0.940·113-s − 0.554·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.987400398017760310895896560809, −6.75229764863286320869533548111, −6.24117366064559688032989386701, −5.55640214918068140655198660256, −4.81835516658297113831283042749, −3.91773099657141212333056164622, −3.15334639735135370079211994334, −2.37461704534527327997602467428, −1.27281607484223001395466480811, 0,
1.27281607484223001395466480811, 2.37461704534527327997602467428, 3.15334639735135370079211994334, 3.91773099657141212333056164622, 4.81835516658297113831283042749, 5.55640214918068140655198660256, 6.24117366064559688032989386701, 6.75229764863286320869533548111, 7.987400398017760310895896560809