| L(s) = 1 | + 3·13-s − 3·17-s + 2·19-s − 3·23-s − 7·29-s − 31-s − 4·37-s + 5·41-s − 3·43-s + 4·47-s + 53-s + 13·59-s + 13·61-s − 4·67-s − 4·71-s − 4·79-s + 7·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.832·13-s − 0.727·17-s + 0.458·19-s − 0.625·23-s − 1.29·29-s − 0.179·31-s − 0.657·37-s + 0.780·41-s − 0.457·43-s + 0.583·47-s + 0.137·53-s + 1.69·59-s + 1.66·61-s − 0.488·67-s − 0.474·71-s − 0.450·79-s + 0.768·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.109487686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109487686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.15959330483212, −12.94179162214651, −12.11459680567321, −11.76384090917557, −11.31663142486499, −10.84854600110694, −10.37945456693334, −9.891055382355760, −9.288122412751987, −8.918006005880095, −8.442863893346578, −7.901591593069147, −7.388323617092120, −6.866483381381948, −6.413791221251053, −5.709920318365750, −5.491608065115411, −4.773087735390572, −4.088634805051331, −3.743279548644042, −3.189713885873875, −2.331539876220151, −1.956370146826596, −1.164138231080156, −0.4401129242819383,
0.4401129242819383, 1.164138231080156, 1.956370146826596, 2.331539876220151, 3.189713885873875, 3.743279548644042, 4.088634805051331, 4.773087735390572, 5.491608065115411, 5.709920318365750, 6.413791221251053, 6.866483381381948, 7.388323617092120, 7.901591593069147, 8.442863893346578, 8.918006005880095, 9.288122412751987, 9.891055382355760, 10.37945456693334, 10.84854600110694, 11.31663142486499, 11.76384090917557, 12.11459680567321, 12.94179162214651, 13.15959330483212
