| L(s) = 1 | − 3-s + 2·7-s + 9-s − 11-s − 2·17-s − 8·19-s − 2·21-s − 2·23-s − 5·25-s − 27-s + 6·29-s + 33-s + 2·37-s + 2·41-s − 4·43-s − 6·47-s − 3·49-s + 2·51-s + 8·53-s + 8·57-s + 8·59-s + 4·61-s + 2·63-s − 12·67-s + 2·69-s − 10·71-s − 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.485·17-s − 1.83·19-s − 0.436·21-s − 0.417·23-s − 25-s − 0.192·27-s + 1.11·29-s + 0.174·33-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s + 0.280·51-s + 1.09·53-s + 1.05·57-s + 1.04·59-s + 0.512·61-s + 0.251·63-s − 1.46·67-s + 0.240·69-s − 1.18·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.475977566058486800212680093202, −8.117442919100094465742462814270, −7.04687809206447612949395852572, −6.32313269200603529321755565165, −5.54998094988422515953569009807, −4.59810532211443779163059075768, −4.06233360349853897957847292507, −2.56902180491320591748454866398, −1.59922369562261816180031479279, 0,
1.59922369562261816180031479279, 2.56902180491320591748454866398, 4.06233360349853897957847292507, 4.59810532211443779163059075768, 5.54998094988422515953569009807, 6.32313269200603529321755565165, 7.04687809206447612949395852572, 8.117442919100094465742462814270, 8.475977566058486800212680093202
