L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 2·11-s − 13-s + 2·15-s − 2·19-s + 4·21-s − 8·23-s − 25-s − 27-s − 2·29-s + 4·31-s − 2·33-s + 8·35-s − 8·37-s + 39-s + 10·41-s − 2·45-s + 8·47-s + 9·49-s − 2·53-s − 4·55-s + 2·57-s − 6·61-s − 4·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.516·15-s − 0.458·19-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s + 1.35·35-s − 1.31·37-s + 0.160·39-s + 1.56·41-s − 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.274·53-s − 0.539·55-s + 0.264·57-s − 0.768·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6581737044\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6581737044\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.95347017800231, −12.45579035397466, −12.34072153421146, −11.85813353980115, −11.32777311420671, −10.85414760417067, −10.30021534593840, −9.798666299011626, −9.554312317625206, −8.882179933483274, −8.401368252001524, −7.728510812539467, −7.386080340735306, −6.762492799635738, −6.369846078511111, −5.932081114076271, −5.474782075080346, −4.565115358607752, −4.207328719388821, −3.630667444103658, −3.362364299675479, −2.424843338807113, −1.941215468216788, −0.8542648614534211, −0.3123697175598013,
0.3123697175598013, 0.8542648614534211, 1.941215468216788, 2.424843338807113, 3.362364299675479, 3.630667444103658, 4.207328719388821, 4.565115358607752, 5.474782075080346, 5.932081114076271, 6.369846078511111, 6.762492799635738, 7.386080340735306, 7.728510812539467, 8.401368252001524, 8.882179933483274, 9.554312317625206, 9.798666299011626, 10.30021534593840, 10.85414760417067, 11.32777311420671, 11.85813353980115, 12.34072153421146, 12.45579035397466, 12.95347017800231