| L(s) = 1 | − 2·3-s − 4·7-s + 9-s + 11-s − 5·13-s + 7·19-s + 8·21-s + 3·23-s + 4·27-s + 3·29-s − 5·31-s − 2·33-s + 4·37-s + 10·39-s + 12·41-s + 5·43-s + 9·49-s − 6·53-s − 14·57-s − 12·59-s − 10·61-s − 4·63-s + 14·67-s − 6·69-s − 3·71-s − 8·73-s − 4·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 1.60·19-s + 1.74·21-s + 0.625·23-s + 0.769·27-s + 0.557·29-s − 0.898·31-s − 0.348·33-s + 0.657·37-s + 1.60·39-s + 1.87·41-s + 0.762·43-s + 9/7·49-s − 0.824·53-s − 1.85·57-s − 1.56·59-s − 1.28·61-s − 0.503·63-s + 1.71·67-s − 0.722·69-s − 0.356·71-s − 0.936·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.63257319890544135439827493203, −7.22202924233692264740330548663, −6.39262870497431449638876163258, −5.87705896156259728281653254280, −5.15576565012033479820483592146, −4.39758809158887942542402516631, −3.24655525711911367891116456446, −2.64926831202420787548361458601, −1.00012265199783966980341878126, 0,
1.00012265199783966980341878126, 2.64926831202420787548361458601, 3.24655525711911367891116456446, 4.39758809158887942542402516631, 5.15576565012033479820483592146, 5.87705896156259728281653254280, 6.39262870497431449638876163258, 7.22202924233692264740330548663, 7.63257319890544135439827493203
