| L(s) = 1 | + 4·2-s − 10·3-s + 16·4-s − 40·6-s − 49·7-s + 64·8-s − 143·9-s − 336·11-s − 160·12-s − 584·13-s − 196·14-s + 256·16-s + 1.45e3·17-s − 572·18-s + 470·19-s + 490·21-s − 1.34e3·22-s + 4.20e3·23-s − 640·24-s − 2.33e3·26-s + 3.86e3·27-s − 784·28-s + 4.86e3·29-s − 7.37e3·31-s + 1.02e3·32-s + 3.36e3·33-s + 5.83e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.641·3-s + 1/2·4-s − 0.453·6-s − 0.377·7-s + 0.353·8-s − 0.588·9-s − 0.837·11-s − 0.320·12-s − 0.958·13-s − 0.267·14-s + 1/4·16-s + 1.22·17-s − 0.416·18-s + 0.298·19-s + 0.242·21-s − 0.592·22-s + 1.65·23-s − 0.226·24-s − 0.677·26-s + 1.01·27-s − 0.188·28-s + 1.07·29-s − 1.37·31-s + 0.176·32-s + 0.537·33-s + 0.865·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.997780092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.997780092\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 10 T + p^{5} T^{2} \) |
| 11 | \( 1 + 336 T + p^{5} T^{2} \) |
| 13 | \( 1 + 584 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 - 470 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 + 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 - 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 - 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 + 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 - 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 - 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 - 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 - 16978 T + p^{5} T^{2} \) |
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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
−10.75095230789914470018139042597, −10.05803388061173802992308703901, −8.786874739777359673634955237011, −7.53000324912367722126541833208, −6.71772842842508203383109369969, −5.35396509392181153631403039775, −5.18935258877630624478602050271, −3.47188924867117868251896705274, −2.51919983149663740387416050336, −0.69616380830731399143520127243,
0.69616380830731399143520127243, 2.51919983149663740387416050336, 3.47188924867117868251896705274, 5.18935258877630624478602050271, 5.35396509392181153631403039775, 6.71772842842508203383109369969, 7.53000324912367722126541833208, 8.786874739777359673634955237011, 10.05803388061173802992308703901, 10.75095230789914470018139042597
