L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 6·11-s + 4·13-s + 16-s + 6·17-s + 4·19-s − 20-s − 6·22-s + 25-s − 4·26-s − 6·29-s + 4·31-s − 32-s − 6·34-s + 8·37-s − 4·38-s + 40-s + 8·43-s + 6·44-s − 50-s + 4·52-s − 6·53-s − 6·55-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.784·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.31·37-s − 0.648·38-s + 0.158·40-s + 1.21·43-s + 0.904·44-s − 0.141·50-s + 0.554·52-s − 0.824·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733127622\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733127622\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.361683788765927099886703120446, −7.68072566687187284194521624333, −7.10097996037311265247610233993, −6.15095560601501333547012713952, −5.76306799226879953191493212190, −4.42900889702114434346086151421, −3.67917655074737520443823096900, −3.02077554855234761959138967953, −1.50258067725827938602043215208, −0.945958207187822378507835712582,
0.945958207187822378507835712582, 1.50258067725827938602043215208, 3.02077554855234761959138967953, 3.67917655074737520443823096900, 4.42900889702114434346086151421, 5.76306799226879953191493212190, 6.15095560601501333547012713952, 7.10097996037311265247610233993, 7.68072566687187284194521624333, 8.361683788765927099886703120446