| L(s) = 1 | + 2·3-s + 9-s + 6·11-s − 6·17-s + 2·19-s − 5·25-s − 4·27-s + 12·33-s + 6·41-s − 10·43-s − 7·49-s − 12·51-s + 4·57-s + 6·59-s − 14·67-s − 2·73-s − 10·75-s − 11·81-s + 18·83-s − 18·89-s + 10·97-s + 6·99-s + 6·107-s + 18·113-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.80·11-s − 1.45·17-s + 0.458·19-s − 25-s − 0.769·27-s + 2.08·33-s + 0.937·41-s − 1.52·43-s − 49-s − 1.68·51-s + 0.529·57-s + 0.781·59-s − 1.71·67-s − 0.234·73-s − 1.15·75-s − 1.22·81-s + 1.97·83-s − 1.90·89-s + 1.01·97-s + 0.603·99-s + 0.580·107-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.781150396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781150396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−11.91995920497978496826321746442, −11.22420990477091317474434780207, −9.739977345448053506810012798737, −9.084650424863502138877874475320, −8.337928305679105143801605101388, −7.14838005963241329022744903293, −6.13287781737238367476060833635, −4.38680180444175422137402538929, −3.38746738060370983088123271545, −1.90996633779170653420430529631,
1.90996633779170653420430529631, 3.38746738060370983088123271545, 4.38680180444175422137402538929, 6.13287781737238367476060833635, 7.14838005963241329022744903293, 8.337928305679105143801605101388, 9.084650424863502138877874475320, 9.739977345448053506810012798737, 11.22420990477091317474434780207, 11.91995920497978496826321746442
