| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s − 44·11-s + 42·13-s + 94·17-s − 36·19-s − 21·21-s − 24·23-s − 27·27-s + 54·29-s − 112·31-s + 132·33-s + 322·37-s − 126·39-s − 22·41-s − 292·43-s − 272·47-s + 49·49-s − 282·51-s + 578·53-s + 108·57-s − 44·59-s − 26·61-s + 63·63-s − 12·67-s + 72·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.896·13-s + 1.34·17-s − 0.434·19-s − 0.218·21-s − 0.217·23-s − 0.192·27-s + 0.345·29-s − 0.648·31-s + 0.696·33-s + 1.43·37-s − 0.517·39-s − 0.0838·41-s − 1.03·43-s − 0.844·47-s + 1/7·49-s − 0.774·51-s + 1.49·53-s + 0.250·57-s − 0.0970·59-s − 0.0545·61-s + 0.125·63-s − 0.0218·67-s + 0.125·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.672868323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672868323\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 24 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 322 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 272 T + p^{3} T^{2} \) |
| 53 | \( 1 - 578 T + p^{3} T^{2} \) |
| 59 | \( 1 + 44 T + p^{3} T^{2} \) |
| 61 | \( 1 + 26 T + p^{3} T^{2} \) |
| 67 | \( 1 + 12 T + p^{3} T^{2} \) |
| 71 | \( 1 + 280 T + p^{3} T^{2} \) |
| 73 | \( 1 + 410 T + p^{3} T^{2} \) |
| 79 | \( 1 + 320 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1252 T + p^{3} T^{2} \) |
| 89 | \( 1 + 38 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1250 T + p^{3} T^{2} \) |
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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
−8.570183794080440146370735293837, −7.978886477052569302036617645104, −7.27109840802113815825751475124, −6.22314462128016838053782585671, −5.58419718994798565951720625741, −4.88125351205549809206160451763, −3.88629182451005082115763452093, −2.88220030380574954091197146972, −1.68355179838325731185657775244, −0.62177367992935995013956624917,
0.62177367992935995013956624917, 1.68355179838325731185657775244, 2.88220030380574954091197146972, 3.88629182451005082115763452093, 4.88125351205549809206160451763, 5.58419718994798565951720625741, 6.22314462128016838053782585671, 7.27109840802113815825751475124, 7.978886477052569302036617645104, 8.570183794080440146370735293837
