| L(s) = 1 | − 4·3-s + 24·7-s − 11·9-s + 44·11-s − 22·13-s − 50·17-s − 44·19-s − 96·21-s − 56·23-s + 152·27-s + 198·29-s + 160·31-s − 176·33-s + 162·37-s + 88·39-s − 198·41-s + 52·43-s + 528·47-s + 233·49-s + 200·51-s + 242·53-s + 176·57-s + 668·59-s + 550·61-s − 264·63-s + 188·67-s + 224·69-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 1.29·7-s − 0.407·9-s + 1.20·11-s − 0.469·13-s − 0.713·17-s − 0.531·19-s − 0.997·21-s − 0.507·23-s + 1.08·27-s + 1.26·29-s + 0.926·31-s − 0.928·33-s + 0.719·37-s + 0.361·39-s − 0.754·41-s + 0.184·43-s + 1.63·47-s + 0.679·49-s + 0.549·51-s + 0.627·53-s + 0.408·57-s + 1.47·59-s + 1.15·61-s − 0.527·63-s + 0.342·67-s + 0.390·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.593793040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593793040\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 198 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 242 T + p^{3} T^{2} \) |
| 59 | \( 1 - 668 T + p^{3} T^{2} \) |
| 61 | \( 1 - 550 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 728 T + p^{3} T^{2} \) |
| 73 | \( 1 + 154 T + p^{3} T^{2} \) |
| 79 | \( 1 - 656 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
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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
−11.02917144489858541020404611391, −10.15566685295665828735539079219, −8.863089382957613941778013360633, −8.212889177451547761589934042420, −6.92223707688319509431382363173, −6.07418720168457991144825688064, −4.96738246788278403594070779344, −4.16952087803522704376667180287, −2.32718986612958753573484937413, −0.882413732014556012127178058365,
0.882413732014556012127178058365, 2.32718986612958753573484937413, 4.16952087803522704376667180287, 4.96738246788278403594070779344, 6.07418720168457991144825688064, 6.92223707688319509431382363173, 8.212889177451547761589934042420, 8.863089382957613941778013360633, 10.15566685295665828735539079219, 11.02917144489858541020404611391
