L(s) = 1 | − 3·3-s + 2·5-s + 6·9-s + 11-s − 7·13-s − 6·15-s + 2·17-s + 8·23-s − 25-s − 9·27-s − 5·29-s − 4·31-s − 3·33-s + 4·37-s + 21·39-s + 4·41-s + 8·43-s + 12·45-s − 2·47-s − 6·51-s − 6·53-s + 2·55-s − 3·59-s + 61-s − 14·65-s − 9·67-s − 24·69-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.894·5-s + 2·9-s + 0.301·11-s − 1.94·13-s − 1.54·15-s + 0.485·17-s + 1.66·23-s − 1/5·25-s − 1.73·27-s − 0.928·29-s − 0.718·31-s − 0.522·33-s + 0.657·37-s + 3.36·39-s + 0.624·41-s + 1.21·43-s + 1.78·45-s − 0.291·47-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 0.390·59-s + 0.128·61-s − 1.73·65-s − 1.09·67-s − 2.88·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.24138112838676071529877740322, −6.67946617283299798598750985856, −5.83912093948941742699162815955, −5.49383997438053223488269666836, −4.85891617558259709174116216939, −4.24676377175376547806194673408, −2.96381076072196429827857946976, −2.00336255656810805561731613280, −1.07696943335018709894615475359, 0,
1.07696943335018709894615475359, 2.00336255656810805561731613280, 2.96381076072196429827857946976, 4.24676377175376547806194673408, 4.85891617558259709174116216939, 5.49383997438053223488269666836, 5.83912093948941742699162815955, 6.67946617283299798598750985856, 7.24138112838676071529877740322