| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 4·11-s − 13-s + 15-s − 17-s + 8·19-s + 4·21-s + 4·23-s + 25-s − 27-s − 2·29-s + 4·33-s + 4·35-s − 6·37-s + 39-s − 6·41-s + 4·43-s − 45-s + 9·49-s + 51-s + 2·53-s + 4·55-s − 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 1.83·19-s + 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.676·35-s − 0.986·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s + 0.140·51-s + 0.274·53-s + 0.539·55-s − 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8117276672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8117276672\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.97078406203894, −12.47837898801235, −12.14643251529277, −11.71313482380091, −11.04560587144070, −10.71370397651252, −10.22201968018115, −9.719713437760151, −9.374505255168980, −8.909597386552043, −8.147088338095364, −7.653474269899525, −7.264315209586517, −6.721739941042349, −6.435670243309541, −5.512750092321981, −5.380066940406516, −4.887597948724612, −4.106875664653715, −3.480522119249609, −3.097571320260148, −2.645751278270458, −1.799765913882409, −0.8665299097698363, −0.3346319668645419,
0.3346319668645419, 0.8665299097698363, 1.799765913882409, 2.645751278270458, 3.097571320260148, 3.480522119249609, 4.106875664653715, 4.887597948724612, 5.380066940406516, 5.512750092321981, 6.435670243309541, 6.721739941042349, 7.264315209586517, 7.653474269899525, 8.147088338095364, 8.909597386552043, 9.374505255168980, 9.719713437760151, 10.22201968018115, 10.71370397651252, 11.04560587144070, 11.71313482380091, 12.14643251529277, 12.47837898801235, 12.97078406203894
