| L(s) = 1 | + 5-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s − 10·37-s − 2·41-s − 4·43-s − 3·45-s + 8·47-s − 2·53-s − 4·55-s − 8·59-s + 14·61-s + 6·65-s + 12·67-s + 16·71-s − 2·73-s + 8·79-s + 9·81-s + 8·83-s − 2·85-s − 10·89-s − 2·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.79·61-s + 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.900·79-s + 81-s + 0.878·83-s − 0.216·85-s − 1.05·89-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784452441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784452441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−18.12659075934542, −17.53823860923922, −17.08312786859915, −16.21594382253087, −15.57209094712168, −15.38005252726752, −14.07366678614862, −13.90751573858575, −13.31256982346756, −12.57905450698390, −11.81010812402662, −11.08364218810912, −10.59209356642135, −9.986591123443433, −8.985245806073316, −8.400283535029584, −8.058922554914412, −6.782480700301191, −6.281362928176271, −5.472549391857356, −4.900546866171145, −3.721799237731026, −2.922578498086050, −2.096637600258421, −0.7596095834913860,
0.7596095834913860, 2.096637600258421, 2.922578498086050, 3.721799237731026, 4.900546866171145, 5.472549391857356, 6.281362928176271, 6.782480700301191, 8.058922554914412, 8.400283535029584, 8.985245806073316, 9.986591123443433, 10.59209356642135, 11.08364218810912, 11.81010812402662, 12.57905450698390, 13.31256982346756, 13.90751573858575, 14.07366678614862, 15.38005252726752, 15.57209094712168, 16.21594382253087, 17.08312786859915, 17.53823860923922, 18.12659075934542
