| L(s) = 1 | + 2·2-s + 8·3-s + 4·4-s − 3·5-s + 16·6-s − 35·7-s + 8·8-s + 37·9-s − 6·10-s + 46·11-s + 32·12-s − 20·13-s − 70·14-s − 24·15-s + 16·16-s − 8·17-s + 74·18-s + 97·19-s − 12·20-s − 280·21-s + 92·22-s − 28·23-s + 64·24-s − 116·25-s − 40·26-s + 80·27-s − 140·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 1/2·4-s − 0.268·5-s + 1.08·6-s − 1.88·7-s + 0.353·8-s + 1.37·9-s − 0.189·10-s + 1.26·11-s + 0.769·12-s − 0.426·13-s − 1.33·14-s − 0.413·15-s + 1/4·16-s − 0.114·17-s + 0.968·18-s + 1.17·19-s − 0.134·20-s − 2.90·21-s + 0.891·22-s − 0.253·23-s + 0.544·24-s − 0.927·25-s − 0.301·26-s + 0.570·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.456590337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.456590337\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 46 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 8 T + p^{3} T^{2} \) |
| 19 | \( 1 - 97 T + p^{3} T^{2} \) |
| 23 | \( 1 + 28 T + p^{3} T^{2} \) |
| 29 | \( 1 - 206 T + p^{3} T^{2} \) |
| 37 | \( 1 - 282 T + p^{3} T^{2} \) |
| 41 | \( 1 - 367 T + p^{3} T^{2} \) |
| 43 | \( 1 - 562 T + p^{3} T^{2} \) |
| 47 | \( 1 + 148 T + p^{3} T^{2} \) |
| 53 | \( 1 - 84 T + p^{3} T^{2} \) |
| 59 | \( 1 + 301 T + p^{3} T^{2} \) |
| 61 | \( 1 - 236 T + p^{3} T^{2} \) |
| 67 | \( 1 - 60 T + p^{3} T^{2} \) |
| 71 | \( 1 - 699 T + p^{3} T^{2} \) |
| 73 | \( 1 - 814 T + p^{3} T^{2} \) |
| 79 | \( 1 + 670 T + p^{3} T^{2} \) |
| 83 | \( 1 - 650 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1566 T + p^{3} T^{2} \) |
| 97 | \( 1 + 615 T + p^{3} T^{2} \) |
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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
−9.034834947508519839063630203995, −7.948230129233789304935582477063, −7.26940299258945945091082117802, −6.52443270865379911678196139565, −5.80465818765950829276294731975, −4.25485496914090638317713521136, −3.78195707837039850144605993236, −2.99690071163460346807815726605, −2.41628020855094404640111324252, −0.907747399641993119843953839689,
0.907747399641993119843953839689, 2.41628020855094404640111324252, 2.99690071163460346807815726605, 3.78195707837039850144605993236, 4.25485496914090638317713521136, 5.80465818765950829276294731975, 6.52443270865379911678196139565, 7.26940299258945945091082117802, 7.948230129233789304935582477063, 9.034834947508519839063630203995
