L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 3·11-s + 5·13-s − 15-s − 6·17-s − 19-s + 21-s + 25-s + 27-s + 5·31-s + 3·33-s − 35-s + 2·37-s + 5·39-s − 6·41-s − 43-s − 45-s − 12·47-s + 49-s − 6·51-s + 6·53-s − 3·55-s − 57-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.38·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 0.898·31-s + 0.522·33-s − 0.169·35-s + 0.328·37-s + 0.800·39-s − 0.937·41-s − 0.152·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 0.404·55-s − 0.132·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.29341187544018, −12.91757778793569, −12.14878913841476, −11.77039900443378, −11.30520148809754, −10.98674516761409, −10.48279942927032, −9.810699311622477, −9.417307551935215, −8.777825269032258, −8.448542353421437, −8.256608774040746, −7.605241591266897, −6.817635180281425, −6.600498303428310, −6.258432271092729, −5.351804189403813, −4.878154418081543, −4.239716467452784, −3.819363961354918, −3.529806250096666, −2.655498323167215, −2.173466569694214, −1.432513043436949, −0.9873229092256220, 0,
0.9873229092256220, 1.432513043436949, 2.173466569694214, 2.655498323167215, 3.529806250096666, 3.819363961354918, 4.239716467452784, 4.878154418081543, 5.351804189403813, 6.258432271092729, 6.600498303428310, 6.817635180281425, 7.605241591266897, 8.256608774040746, 8.448542353421437, 8.777825269032258, 9.417307551935215, 9.810699311622477, 10.48279942927032, 10.98674516761409, 11.30520148809754, 11.77039900443378, 12.14878913841476, 12.91757778793569, 13.29341187544018