L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s − 6·13-s + 16-s − 4·17-s − 6·19-s − 20-s + 4·22-s + 25-s + 6·26-s − 6·29-s + 4·31-s − 32-s + 4·34-s + 8·37-s + 6·38-s + 40-s − 10·41-s − 2·43-s − 4·44-s − 10·47-s − 50-s − 6·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 1.17·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s + 1.31·37-s + 0.973·38-s + 0.158·40-s − 1.56·41-s − 0.304·43-s − 0.603·44-s − 1.45·47-s − 0.141·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4266182995\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4266182995\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.218003904598523798299424809353, −7.83261236693941610700137644746, −6.98653087432088301263168730843, −6.47635875232034878216320965993, −5.29570705758244311671020673600, −4.74386920329066003916598022870, −3.75063151244410675840057189340, −2.52407845957186438364572976596, −2.14920177633157734626583490017, −0.37625829394764788591833939327,
0.37625829394764788591833939327, 2.14920177633157734626583490017, 2.52407845957186438364572976596, 3.75063151244410675840057189340, 4.74386920329066003916598022870, 5.29570705758244311671020673600, 6.47635875232034878216320965993, 6.98653087432088301263168730843, 7.83261236693941610700137644746, 8.218003904598523798299424809353