L(s) = 1 | − 3-s − 2·4-s − 7-s + 9-s + 2·12-s + 2·13-s + 4·16-s + 6·17-s + 7·19-s + 21-s + 6·23-s − 27-s + 2·28-s − 31-s − 2·36-s + 7·37-s − 2·39-s + 6·41-s + 8·43-s − 4·48-s − 6·49-s − 6·51-s − 4·52-s + 6·53-s − 7·57-s − 12·59-s + 61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 0.577·12-s + 0.554·13-s + 16-s + 1.45·17-s + 1.60·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.377·28-s − 0.179·31-s − 1/3·36-s + 1.15·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.577·48-s − 6/7·49-s − 0.840·51-s − 0.554·52-s + 0.824·53-s − 0.927·57-s − 1.56·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.541823506\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.541823506\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.57429178589206752599825058577, −7.26171901265395654422771855257, −6.04510595268873663188530754064, −5.71649178853268531344450376349, −5.02152432805084214325173207341, −4.31963574436031364627776411170, −3.45739773570944288331043111385, −2.93093200895283056957551589461, −1.26269753115414327598647616864, −0.74876154913487424076601045851,
0.74876154913487424076601045851, 1.26269753115414327598647616864, 2.93093200895283056957551589461, 3.45739773570944288331043111385, 4.31963574436031364627776411170, 5.02152432805084214325173207341, 5.71649178853268531344450376349, 6.04510595268873663188530754064, 7.26171901265395654422771855257, 7.57429178589206752599825058577