L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·11-s + 13-s − 15-s + 17-s − 4·19-s − 21-s − 8·23-s + 25-s − 27-s − 2·29-s − 4·33-s + 35-s + 6·37-s − 39-s + 10·41-s + 4·43-s + 45-s + 49-s − 51-s − 10·53-s + 4·55-s + 4·57-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.140·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.813933462\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.813933462\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.42536145394794, −12.02569769069637, −11.64223067919237, −11.08123196133471, −10.78684494490655, −10.32527605727014, −9.751988455861304, −9.261252076313470, −9.104806892082676, −8.349834745617292, −7.771905755524037, −7.647281732731432, −6.744947542183586, −6.332023521022173, −6.112892619298411, −5.649784049090289, −4.978191214446445, −4.468001954646831, −3.973011881428919, −3.698392045918998, −2.787556221245564, −2.120900318430339, −1.752610185417086, −1.066256697358743, −0.4921765066658509,
0.4921765066658509, 1.066256697358743, 1.752610185417086, 2.120900318430339, 2.787556221245564, 3.698392045918998, 3.973011881428919, 4.468001954646831, 4.978191214446445, 5.649784049090289, 6.112892619298411, 6.332023521022173, 6.744947542183586, 7.647281732731432, 7.771905755524037, 8.349834745617292, 9.104806892082676, 9.261252076313470, 9.751988455861304, 10.32527605727014, 10.78684494490655, 11.08123196133471, 11.64223067919237, 12.02569769069637, 12.42536145394794