| L(s) = 1 | − 2·4-s + 5-s − 7-s + 13-s + 4·16-s − 6·17-s + 2·19-s − 2·20-s + 3·23-s + 25-s + 2·28-s − 4·31-s − 35-s − 7·37-s + 3·41-s − 10·43-s − 6·47-s − 6·49-s − 2·52-s + 9·53-s + 9·59-s − 10·61-s − 8·64-s + 65-s + 5·67-s + 12·68-s − 15·71-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s + 0.277·13-s + 16-s − 1.45·17-s + 0.458·19-s − 0.447·20-s + 0.625·23-s + 1/5·25-s + 0.377·28-s − 0.718·31-s − 0.169·35-s − 1.15·37-s + 0.468·41-s − 1.52·43-s − 0.875·47-s − 6/7·49-s − 0.277·52-s + 1.23·53-s + 1.17·59-s − 1.28·61-s − 64-s + 0.124·65-s + 0.610·67-s + 1.45·68-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.888635785842699180841629185842, −8.439851655181107560137829138095, −7.25784866905101757102236573749, −6.49919525151324242085523605567, −5.53851423966891790044824368427, −4.82999517464359753479089320052, −3.90452398007734281695986656605, −2.95407140568710005546864602227, −1.56264258069941371391730839809, 0,
1.56264258069941371391730839809, 2.95407140568710005546864602227, 3.90452398007734281695986656605, 4.82999517464359753479089320052, 5.53851423966891790044824368427, 6.49919525151324242085523605567, 7.25784866905101757102236573749, 8.439851655181107560137829138095, 8.888635785842699180841629185842
