L(s) = 1 | − 4i·3-s + 2i·5-s − 24·7-s + 11·9-s + 44i·11-s + 22i·13-s + 8·15-s + 50·17-s + 44i·19-s + 96i·21-s + 56·23-s + 121·25-s − 152i·27-s + 198i·29-s − 160·31-s + ⋯ |
L(s) = 1 | − 0.769i·3-s + 0.178i·5-s − 1.29·7-s + 0.407·9-s + 1.20i·11-s + 0.469i·13-s + 0.137·15-s + 0.713·17-s + 0.531i·19-s + 0.997i·21-s + 0.507·23-s + 0.967·25-s − 1.08i·27-s + 1.26i·29-s − 0.926·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.380062326\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.380062326\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 4iT - 27T^{2} \) |
| 5 | \( 1 - 2iT - 125T^{2} \) |
| 7 | \( 1 + 24T + 343T^{2} \) |
| 11 | \( 1 - 44iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 50T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 56T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 160T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 198T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 528T + 1.03e5T^{2} \) |
| 53 | \( 1 - 242iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 668iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 550iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 188iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 728T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154T + 3.89e5T^{2} \) |
| 79 | \( 1 + 656T + 4.93e5T^{2} \) |
| 83 | \( 1 - 236iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 714T + 7.04e5T^{2} \) |
| 97 | \( 1 + 478T + 9.12e5T^{2} \) |
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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
−12.11612707463161875523685853223, −10.57723738449281527308274480460, −9.820017050472189887352634511801, −8.886174068167655909965859509063, −7.24457423617757938796820382894, −7.05410609786882479615949649298, −5.81526475236632263694665973477, −4.26080451446738959082772263379, −2.86153153938665675655990324674, −1.35557532313524436370529560537,
0.59803423341275942997376131769, 2.98632984558379587973162317561, 3.87861137604062335193575549335, 5.28002614422809057896061428657, 6.28950060603509690117392218994, 7.48749552587012996451964213837, 8.847881655349570779554331129719, 9.550780307567013017507173911884, 10.42538900663903027115445844205, 11.22764714299319912895341675107