| L(s) = 1 | − 3-s + 2·7-s + 9-s − 13-s − 6·17-s − 2·19-s − 2·21-s − 5·25-s − 27-s + 6·29-s + 2·31-s − 2·37-s + 39-s − 12·41-s + 4·43-s − 3·49-s + 6·51-s − 6·53-s + 2·57-s − 12·59-s − 2·61-s + 2·63-s + 10·67-s + 12·71-s + 14·73-s + 5·75-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.436·21-s − 25-s − 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.328·37-s + 0.160·39-s − 1.87·41-s + 0.609·43-s − 3/7·49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s − 1.56·59-s − 0.256·61-s + 0.251·63-s + 1.22·67-s + 1.42·71-s + 1.63·73-s + 0.577·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.397749324163862668074879881731, −7.903021947901095999330816312888, −6.77588051345062669609369199457, −6.38841228560375566367348698112, −5.23796721936294605278821689684, −4.69511531925842197635869476162, −3.86363777727094941125777069865, −2.49852565048742461389380956784, −1.54822382936130812607770538657, 0,
1.54822382936130812607770538657, 2.49852565048742461389380956784, 3.86363777727094941125777069865, 4.69511531925842197635869476162, 5.23796721936294605278821689684, 6.38841228560375566367348698112, 6.77588051345062669609369199457, 7.903021947901095999330816312888, 8.397749324163862668074879881731
