L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 3·11-s − 4·13-s + 2·14-s + 16-s + 3·17-s + 5·19-s + 3·22-s − 6·23-s − 4·26-s + 2·28-s + 2·31-s + 32-s + 3·34-s + 2·37-s + 5·38-s + 3·41-s − 4·43-s + 3·44-s − 6·46-s − 12·47-s − 3·49-s − 4·52-s − 6·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.904·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s + 0.639·22-s − 1.25·23-s − 0.784·26-s + 0.377·28-s + 0.359·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s + 0.811·38-s + 0.468·41-s − 0.609·43-s + 0.452·44-s − 0.884·46-s − 1.75·47-s − 3/7·49-s − 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.305898567\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305898567\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−11.52023790049563233004214565881, −10.18789957492033670124802464882, −9.481094579683641415186193216160, −8.114692805267148178564014319106, −7.40487753845237746133032109213, −6.28477478517999960844504626470, −5.23380204015925208976616676091, −4.36763710614752510700235050554, −3.13165277251593234405465211013, −1.63862417385241407780156087735,
1.63862417385241407780156087735, 3.13165277251593234405465211013, 4.36763710614752510700235050554, 5.23380204015925208976616676091, 6.28477478517999960844504626470, 7.40487753845237746133032109213, 8.114692805267148178564014319106, 9.481094579683641415186193216160, 10.18789957492033670124802464882, 11.52023790049563233004214565881