L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 6·11-s − 12-s − 2·13-s − 16-s + 4·17-s + 18-s + 6·19-s − 6·22-s − 3·24-s − 2·26-s + 27-s − 2·29-s + 10·31-s + 5·32-s − 6·33-s + 4·34-s − 36-s + 4·37-s + 6·38-s − 2·39-s − 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.554·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.37·19-s − 1.27·22-s − 0.612·24-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 1.79·31-s + 0.883·32-s − 1.04·33-s + 0.685·34-s − 1/6·36-s + 0.657·37-s + 0.973·38-s − 0.320·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.355304227\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.355304227\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.245873608153951043349925317243, −7.976741612740113433163431746666, −7.18600650759489131977738022987, −6.06490562142255136482234133151, −5.20476390495317100873412386991, −4.93935473109955635441904702605, −3.83469354318778825175338832285, −3.01958613902430109855645796176, −2.47409506150476348759674637901, −0.77490080633513895615658874599,
0.77490080633513895615658874599, 2.47409506150476348759674637901, 3.01958613902430109855645796176, 3.83469354318778825175338832285, 4.93935473109955635441904702605, 5.20476390495317100873412386991, 6.06490562142255136482234133151, 7.18600650759489131977738022987, 7.976741612740113433163431746666, 8.245873608153951043349925317243