| L(s) = 1 | − 3-s − 4·7-s + 9-s − 2·11-s − 4·13-s − 17-s + 4·19-s + 4·21-s − 8·23-s − 27-s + 8·29-s + 2·33-s + 2·37-s + 4·39-s − 4·41-s + 6·43-s − 12·47-s + 9·49-s + 51-s − 14·53-s − 4·57-s + 2·61-s − 4·63-s + 2·67-s + 8·69-s − 14·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.242·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 1.48·29-s + 0.348·33-s + 0.328·37-s + 0.640·39-s − 0.624·41-s + 0.914·43-s − 1.75·47-s + 9/7·49-s + 0.140·51-s − 1.92·53-s − 0.529·57-s + 0.256·61-s − 0.503·63-s + 0.244·67-s + 0.963·69-s − 1.66·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2247590175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2247590175\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.74204111044190, −15.40121817556934, −14.35860461554662, −14.08966941878661, −13.32333537108345, −12.74723077228054, −12.50180380993533, −11.76012347883085, −11.44734186008884, −10.35433152129622, −10.12279999553137, −9.722503101562773, −9.112378762261052, −8.208574120059607, −7.642721122702005, −7.032792475571812, −6.356844325615584, −6.026421099815404, −5.192973525779386, −4.646266511578271, −3.857767917107552, −3.001942483920584, −2.581867221389900, −1.454822480901571, −0.1995120269963112,
0.1995120269963112, 1.454822480901571, 2.581867221389900, 3.001942483920584, 3.857767917107552, 4.646266511578271, 5.192973525779386, 6.026421099815404, 6.356844325615584, 7.032792475571812, 7.642721122702005, 8.208574120059607, 9.112378762261052, 9.722503101562773, 10.12279999553137, 10.35433152129622, 11.44734186008884, 11.76012347883085, 12.50180380993533, 12.74723077228054, 13.32333537108345, 14.08966941878661, 14.35860461554662, 15.40121817556934, 15.74204111044190
