L(s) = 1 | − 7-s − 3·9-s − 11-s + 2·13-s + 4·17-s + 2·19-s − 5·25-s − 2·29-s + 10·31-s − 10·37-s − 4·41-s − 8·43-s + 6·47-s + 49-s + 2·53-s + 4·59-s + 10·61-s + 3·63-s − 12·67-s + 4·71-s + 4·73-s + 77-s + 9·81-s − 14·83-s − 14·89-s − 2·91-s + 2·97-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s − 0.301·11-s + 0.554·13-s + 0.970·17-s + 0.458·19-s − 25-s − 0.371·29-s + 1.79·31-s − 1.64·37-s − 0.624·41-s − 1.21·43-s + 0.875·47-s + 1/7·49-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.377·63-s − 1.46·67-s + 0.474·71-s + 0.468·73-s + 0.113·77-s + 81-s − 1.53·83-s − 1.48·89-s − 0.209·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.110497411070536479244874590455, −7.14736803834861553458798560954, −6.42145514289455283738783938047, −5.61158295850811917204228613392, −5.21647672363043619820980044159, −3.98612394874942882803971095617, −3.28811494948009498877743064219, −2.55034194243224838160377458120, −1.31637782376016475054643556876, 0,
1.31637782376016475054643556876, 2.55034194243224838160377458120, 3.28811494948009498877743064219, 3.98612394874942882803971095617, 5.21647672363043619820980044159, 5.61158295850811917204228613392, 6.42145514289455283738783938047, 7.14736803834861553458798560954, 8.110497411070536479244874590455