| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 17-s − 4·19-s − 21-s − 8·23-s − 25-s + 27-s + 6·29-s − 4·33-s + 2·35-s − 2·37-s − 2·39-s + 10·41-s + 4·43-s − 2·45-s + 49-s + 51-s + 6·53-s + 8·55-s − 4·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.242·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.338·35-s − 0.328·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.140·51-s + 0.824·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.173126776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.173126776\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.040775186303147997783269223007, −7.65318267549180248234850998911, −6.85789587263237432564868315197, −6.01261730651409420828008874179, −5.17090680208676282944058226105, −4.23426001188259797781555407482, −3.78123059627148327231895163758, −2.70744940692180522192093941737, −2.18240029644083449001087303092, −0.52668238867765794753070779314,
0.52668238867765794753070779314, 2.18240029644083449001087303092, 2.70744940692180522192093941737, 3.78123059627148327231895163758, 4.23426001188259797781555407482, 5.17090680208676282944058226105, 6.01261730651409420828008874179, 6.85789587263237432564868315197, 7.65318267549180248234850998911, 8.040775186303147997783269223007
