L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s + 4·13-s − 4·19-s − 9·21-s + 8·23-s − 9·27-s − 6·29-s − 2·31-s − 8·37-s − 12·39-s − 5·41-s + 5·43-s + 3·47-s + 2·49-s + 4·53-s + 12·57-s + 2·59-s + 11·61-s + 18·63-s − 13·67-s − 24·69-s + 2·71-s + 8·73-s + 10·79-s + 9·81-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s + 1.10·13-s − 0.917·19-s − 1.96·21-s + 1.66·23-s − 1.73·27-s − 1.11·29-s − 0.359·31-s − 1.31·37-s − 1.92·39-s − 0.780·41-s + 0.762·43-s + 0.437·47-s + 2/7·49-s + 0.549·53-s + 1.58·57-s + 0.260·59-s + 1.40·61-s + 2.26·63-s − 1.58·67-s − 2.88·69-s + 0.237·71-s + 0.936·73-s + 1.12·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.692343904\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.692343904\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.02998375949769, −12.52310699952105, −12.07641254211791, −11.50158819391248, −11.22098884061772, −10.87869643661402, −10.52593833051860, −10.10094675404021, −9.198097294052198, −8.879772711097914, −8.363811845874144, −7.691126099245491, −7.217745259646697, −6.690165794121929, −6.343929414296256, −5.615487722856980, −5.347818175466820, −4.925024975430519, −4.339828140162806, −3.845807677393413, −3.226532496824308, −2.123279134475401, −1.689462459756239, −1.031129080430377, −0.4896364332832257,
0.4896364332832257, 1.031129080430377, 1.689462459756239, 2.123279134475401, 3.226532496824308, 3.845807677393413, 4.339828140162806, 4.925024975430519, 5.347818175466820, 5.615487722856980, 6.343929414296256, 6.690165794121929, 7.217745259646697, 7.691126099245491, 8.363811845874144, 8.879772711097914, 9.198097294052198, 10.10094675404021, 10.52593833051860, 10.87869643661402, 11.22098884061772, 11.50158819391248, 12.07641254211791, 12.52310699952105, 13.02998375949769