L(s) = 1 | − 3-s − 4·7-s + 9-s + 6·11-s − 13-s − 4·17-s − 2·19-s + 4·21-s − 6·23-s − 27-s + 10·29-s + 4·31-s − 6·33-s + 6·37-s + 39-s + 10·41-s + 8·47-s + 9·49-s + 4·51-s + 6·53-s + 2·57-s + 6·59-s + 6·61-s − 4·63-s + 12·67-s + 6·69-s + 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s − 0.970·17-s − 0.458·19-s + 0.872·21-s − 1.25·23-s − 0.192·27-s + 1.85·29-s + 0.718·31-s − 1.04·33-s + 0.986·37-s + 0.160·39-s + 1.56·41-s + 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s + 0.264·57-s + 0.781·59-s + 0.768·61-s − 0.503·63-s + 1.46·67-s + 0.722·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.745666808\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.745666808\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.16169391684937, −13.83829551660266, −13.12236946992618, −12.73103756592559, −12.17565923895748, −11.84490768035866, −11.36992785466911, −10.64691212348697, −10.10774769248526, −9.754350466972725, −9.176469168591606, −8.789445688881881, −8.103565114304675, −7.301096972486749, −6.696984959638245, −6.318878171949908, −6.206024924090149, −5.352964655037278, −4.366430663766284, −4.162374156248890, −3.607090534911123, −2.636083728199953, −2.226807294746174, −1.038727100364010, −0.5585331416724146,
0.5585331416724146, 1.038727100364010, 2.226807294746174, 2.636083728199953, 3.607090534911123, 4.162374156248890, 4.366430663766284, 5.352964655037278, 6.206024924090149, 6.318878171949908, 6.696984959638245, 7.301096972486749, 8.103565114304675, 8.789445688881881, 9.176469168591606, 9.754350466972725, 10.10774769248526, 10.64691212348697, 11.36992785466911, 11.84490768035866, 12.17565923895748, 12.73103756592559, 13.12236946992618, 13.83829551660266, 14.16169391684937