L(s) = 1 | − 3-s − 5-s + 9-s − 13-s + 15-s + 2·17-s − 4·19-s + 4·23-s + 25-s − 27-s − 6·29-s + 8·31-s − 6·37-s + 39-s − 2·41-s + 4·43-s − 45-s − 7·49-s − 2·51-s + 6·53-s + 4·57-s − 2·61-s + 65-s + 8·67-s − 4·69-s + 6·73-s − 75-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.256·61-s + 0.124·65-s + 0.977·67-s − 0.481·69-s + 0.702·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.63287802668294287424964171629, −6.94679715592600138420938405730, −6.34869310339064223681880997863, −5.49384900552159011989007079619, −4.85698928413128374035397572891, −4.10407484786934996910917478167, −3.29953796133022934741614363303, −2.29245778312913955468138393628, −1.15941429236263001794663313876, 0,
1.15941429236263001794663313876, 2.29245778312913955468138393628, 3.29953796133022934741614363303, 4.10407484786934996910917478167, 4.85698928413128374035397572891, 5.49384900552159011989007079619, 6.34869310339064223681880997863, 6.94679715592600138420938405730, 7.63287802668294287424964171629