| L(s) = 1 | − 8·3-s + 12·5-s − 32·7-s + 37·9-s + 8·11-s − 20·13-s − 96·15-s − 98·17-s + 88·19-s + 256·21-s + 32·23-s + 19·25-s − 80·27-s + 172·29-s + 256·31-s − 64·33-s − 384·35-s + 92·37-s + 160·39-s + 102·41-s + 296·43-s + 444·45-s + 320·47-s + 681·49-s + 784·51-s + 76·53-s + 96·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 1.07·5-s − 1.72·7-s + 1.37·9-s + 0.219·11-s − 0.426·13-s − 1.65·15-s − 1.39·17-s + 1.06·19-s + 2.66·21-s + 0.290·23-s + 0.151·25-s − 0.570·27-s + 1.10·29-s + 1.48·31-s − 0.337·33-s − 1.85·35-s + 0.408·37-s + 0.656·39-s + 0.388·41-s + 1.04·43-s + 1.47·45-s + 0.993·47-s + 1.98·49-s + 2.15·51-s + 0.196·53-s + 0.235·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8786818795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8786818795\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 98 T + p^{3} T^{2} \) |
| 19 | \( 1 - 88 T + p^{3} T^{2} \) |
| 23 | \( 1 - 32 T + p^{3} T^{2} \) |
| 29 | \( 1 - 172 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 - 92 T + p^{3} T^{2} \) |
| 41 | \( 1 - 102 T + p^{3} T^{2} \) |
| 43 | \( 1 - 296 T + p^{3} T^{2} \) |
| 47 | \( 1 - 320 T + p^{3} T^{2} \) |
| 53 | \( 1 - 76 T + p^{3} T^{2} \) |
| 59 | \( 1 + 408 T + p^{3} T^{2} \) |
| 61 | \( 1 - 636 T + p^{3} T^{2} \) |
| 67 | \( 1 + 552 T + p^{3} T^{2} \) |
| 71 | \( 1 + 416 T + p^{3} T^{2} \) |
| 73 | \( 1 - 138 T + p^{3} T^{2} \) |
| 79 | \( 1 - 64 T + p^{3} T^{2} \) |
| 83 | \( 1 + 392 T + p^{3} T^{2} \) |
| 89 | \( 1 + 582 T + p^{3} T^{2} \) |
| 97 | \( 1 - 238 T + p^{3} T^{2} \) |
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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
−11.64230394274837270651568835716, −10.49456656517206369448463674114, −9.873364293942083345353634661190, −9.096170267354020248336230362697, −7.05359942711422536857798379532, −6.33110976002528288935016550073, −5.75653707558652282163136496634, −4.51095361989727078837816582791, −2.70502047553075965569553815751, −0.71497672314750309377197420862,
0.71497672314750309377197420862, 2.70502047553075965569553815751, 4.51095361989727078837816582791, 5.75653707558652282163136496634, 6.33110976002528288935016550073, 7.05359942711422536857798379532, 9.096170267354020248336230362697, 9.873364293942083345353634661190, 10.49456656517206369448463674114, 11.64230394274837270651568835716
