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
\(L(1)\) |
\(\approx\) |
\(0.7840189550\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7840189550\) |
\(L(\frac{3}{2})\) |
|
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
−14.09363921654168, −13.68776899820697, −13.24846572573878, −12.54892410329767, −12.38440292523640, −12.00595932137533, −11.01859460714406, −10.69608537772919, −10.16064737536441, −9.828435813859468, −9.350089942997679, −8.542465098584158, −7.939770275789257, −7.545117791420853, −6.897500772457154, −6.287160384578027, −5.841472856013184, −5.361183320148293, −4.811866873876481, −3.933225902550163, −3.417499542900161, −2.761567122136218, −2.264132629405907, −1.086415116644853, −0.3504523398205744,
0.3504523398205744, 1.086415116644853, 2.264132629405907, 2.761567122136218, 3.417499542900161, 3.933225902550163, 4.811866873876481, 5.361183320148293, 5.841472856013184, 6.287160384578027, 6.897500772457154, 7.545117791420853, 7.939770275789257, 8.542465098584158, 9.350089942997679, 9.828435813859468, 10.16064737536441, 10.69608537772919, 11.01859460714406, 12.00595932137533, 12.38440292523640, 12.54892410329767, 13.24846572573878, 13.68776899820697, 14.09363921654168