| L(s) = 1 | − 3-s + 7-s − 2·9-s + 11-s + 2·13-s + 5·17-s − 7·19-s − 21-s + 6·23-s + 5·27-s − 29-s − 5·31-s − 33-s − 11·37-s − 2·39-s + 2·41-s − 4·43-s − 6·47-s − 6·49-s − 5·51-s + 53-s + 7·57-s − 10·59-s + 5·61-s − 2·63-s + 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.554·13-s + 1.21·17-s − 1.60·19-s − 0.218·21-s + 1.25·23-s + 0.962·27-s − 0.185·29-s − 0.898·31-s − 0.174·33-s − 1.80·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s − 0.700·51-s + 0.137·53-s + 0.927·57-s − 1.30·59-s + 0.640·61-s − 0.251·63-s + 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.33563938941148056826061674084, −6.59295231671976519886053935963, −6.09211837337601353679537134214, −5.24424507160539686377725717788, −4.89413338908764577870067948724, −3.76126076913644485330927767443, −3.21767878271010981425020671998, −2.08159258772918235229012105266, −1.18706386338281584187335642323, 0,
1.18706386338281584187335642323, 2.08159258772918235229012105266, 3.21767878271010981425020671998, 3.76126076913644485330927767443, 4.89413338908764577870067948724, 5.24424507160539686377725717788, 6.09211837337601353679537134214, 6.59295231671976519886053935963, 7.33563938941148056826061674084
