| L(s) = 1 | − 3-s + 5-s − 2·9-s + 2·11-s − 15-s − 4·17-s + 2·19-s + 23-s + 25-s + 5·27-s + 9·29-s − 4·31-s − 2·33-s + 4·37-s − 41-s + 9·43-s − 2·45-s + 4·51-s − 10·53-s + 2·55-s − 2·57-s + 10·59-s − 9·61-s + 5·67-s − 69-s + 14·71-s − 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.603·11-s − 0.258·15-s − 0.970·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s − 0.718·31-s − 0.348·33-s + 0.657·37-s − 0.156·41-s + 1.37·43-s − 0.298·45-s + 0.560·51-s − 1.37·53-s + 0.269·55-s − 0.264·57-s + 1.30·59-s − 1.15·61-s + 0.610·67-s − 0.120·69-s + 1.66·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426423692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426423692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.126447015637667693088500493602, −8.577853488051574153517458260879, −7.55197449397548355810730048611, −6.54726694118631653414664331561, −6.13381189947026461337417746669, −5.18289036336620639563064974665, −4.44525519825866077450144641008, −3.23526529985027086432912393956, −2.22050794547907166274142371283, −0.827091807844068581217059306961,
0.827091807844068581217059306961, 2.22050794547907166274142371283, 3.23526529985027086432912393956, 4.44525519825866077450144641008, 5.18289036336620639563064974665, 6.13381189947026461337417746669, 6.54726694118631653414664331561, 7.55197449397548355810730048611, 8.577853488051574153517458260879, 9.126447015637667693088500493602
