| L(s) = 1 | − 2·2-s + 9·3-s + 4·4-s − 18·6-s + 5·7-s − 8·8-s + 54·9-s + 11·11-s + 36·12-s + 36·13-s − 10·14-s + 16·16-s − 17·17-s − 108·18-s + 41·19-s + 45·21-s − 22·22-s − 44·23-s − 72·24-s − 72·26-s + 243·27-s + 20·28-s + 285·29-s − 323·31-s − 32·32-s + 99·33-s + 34·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.269·7-s − 0.353·8-s + 2·9-s + 0.301·11-s + 0.866·12-s + 0.768·13-s − 0.190·14-s + 1/4·16-s − 0.242·17-s − 1.41·18-s + 0.495·19-s + 0.467·21-s − 0.213·22-s − 0.398·23-s − 0.612·24-s − 0.543·26-s + 1.73·27-s + 0.134·28-s + 1.82·29-s − 1.87·31-s − 0.176·32-s + 0.522·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.044932116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.044932116\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 13 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + p T + p^{3} T^{2} \) |
| 19 | \( 1 - 41 T + p^{3} T^{2} \) |
| 23 | \( 1 + 44 T + p^{3} T^{2} \) |
| 29 | \( 1 - 285 T + p^{3} T^{2} \) |
| 31 | \( 1 + 323 T + p^{3} T^{2} \) |
| 37 | \( 1 - 29 T + p^{3} T^{2} \) |
| 41 | \( 1 - 208 T + p^{3} T^{2} \) |
| 43 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 336 T + p^{3} T^{2} \) |
| 53 | \( 1 - 725 T + p^{3} T^{2} \) |
| 59 | \( 1 + 648 T + p^{3} T^{2} \) |
| 61 | \( 1 + 565 T + p^{3} T^{2} \) |
| 67 | \( 1 + 748 T + p^{3} T^{2} \) |
| 71 | \( 1 + 265 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 T + p^{3} T^{2} \) |
| 83 | \( 1 + 708 T + p^{3} T^{2} \) |
| 89 | \( 1 - 137 T + p^{3} T^{2} \) |
| 97 | \( 1 - 44 T + p^{3} T^{2} \) |
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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
−10.15283099625317010274292571112, −9.077303289631096885403297771596, −8.862177161011463561348875830076, −7.84271928476332624326602902037, −7.27246927944236128083754679105, −6.03272539093515742649691276540, −4.34317404820961941751475694203, −3.33635158344998779968628442237, −2.30681791509935734057544406926, −1.20278049380753334928631160638,
1.20278049380753334928631160638, 2.30681791509935734057544406926, 3.33635158344998779968628442237, 4.34317404820961941751475694203, 6.03272539093515742649691276540, 7.27246927944236128083754679105, 7.84271928476332624326602902037, 8.862177161011463561348875830076, 9.077303289631096885403297771596, 10.15283099625317010274292571112
