L(s) = 1 | − 2·3-s + 7-s + 9-s − 2·11-s − 5·13-s + 5·17-s + 4·19-s − 2·21-s − 23-s + 4·27-s + 2·29-s + 4·33-s + 9·37-s + 10·39-s + 4·43-s + 47-s + 49-s − 10·51-s − 5·53-s − 8·57-s + 12·59-s + 4·61-s + 63-s − 4·67-s + 2·69-s − 2·71-s + 2·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 1.21·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s + 0.371·29-s + 0.696·33-s + 1.47·37-s + 1.60·39-s + 0.609·43-s + 0.145·47-s + 1/7·49-s − 1.40·51-s − 0.686·53-s − 1.05·57-s + 1.56·59-s + 0.512·61-s + 0.125·63-s − 0.488·67-s + 0.240·69-s − 0.237·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.445282464\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.445282464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.45902747700621, −13.72448948638372, −13.11670971236557, −12.54071821693744, −12.09310493460612, −11.79730499237113, −11.29604497530846, −10.68120576222467, −10.28352186023504, −9.665742583848707, −9.421303080285737, −8.391913956218177, −7.972015083052386, −7.409212184356695, −7.023362581919559, −6.224675543483602, −5.639116233358934, −5.356888684182870, −4.773959793262904, −4.312431147733085, −3.317078113358673, −2.751299498741644, −2.068310877787747, −1.027586841866586, −0.5253158872978368,
0.5253158872978368, 1.027586841866586, 2.068310877787747, 2.751299498741644, 3.317078113358673, 4.312431147733085, 4.773959793262904, 5.356888684182870, 5.639116233358934, 6.224675543483602, 7.023362581919559, 7.409212184356695, 7.972015083052386, 8.391913956218177, 9.421303080285737, 9.665742583848707, 10.28352186023504, 10.68120576222467, 11.29604497530846, 11.79730499237113, 12.09310493460612, 12.54071821693744, 13.11670971236557, 13.72448948638372, 14.45902747700621