L(s) = 1 | + 2·5-s + 2·11-s + 13-s + 4·19-s + 4·23-s − 25-s − 8·29-s + 8·31-s + 2·37-s + 6·41-s − 4·43-s + 6·47-s − 7·49-s − 4·53-s + 4·55-s + 6·59-s + 2·61-s + 2·65-s + 4·67-s + 6·71-s − 2·73-s − 16·79-s + 2·83-s + 10·89-s + 8·95-s − 2·97-s + 8·101-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.603·11-s + 0.277·13-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.875·47-s − 49-s − 0.549·53-s + 0.539·55-s + 0.781·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + 0.712·71-s − 0.234·73-s − 1.80·79-s + 0.219·83-s + 1.05·89-s + 0.820·95-s − 0.203·97-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.526748104\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.526748104\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.609685173226179453665442258483, −7.74652499317951017964613601700, −6.98814297626254924193548047807, −6.20778793559778146432602907404, −5.62349404399732664506023989791, −4.81281047705604358963746290044, −3.84357586762384102764374571608, −2.94793843774524120975754866963, −1.93838116414026850576262121415, −0.979418705281808469159785692374,
0.979418705281808469159785692374, 1.93838116414026850576262121415, 2.94793843774524120975754866963, 3.84357586762384102764374571608, 4.81281047705604358963746290044, 5.62349404399732664506023989791, 6.20778793559778146432602907404, 6.98814297626254924193548047807, 7.74652499317951017964613601700, 8.609685173226179453665442258483