L(s) = 1 | − 2-s − 3-s + 4-s + 1.18·5-s + 6-s + 0.532·7-s − 8-s + 9-s − 1.18·10-s + 1.87·11-s − 12-s − 3.87·13-s − 0.532·14-s − 1.18·15-s + 16-s + 1.16·17-s − 18-s + 1.18·20-s − 0.532·21-s − 1.87·22-s − 6.70·23-s + 24-s − 3.59·25-s + 3.87·26-s − 27-s + 0.532·28-s − 4.02·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.529·5-s + 0.408·6-s + 0.201·7-s − 0.353·8-s + 0.333·9-s − 0.374·10-s + 0.566·11-s − 0.288·12-s − 1.07·13-s − 0.142·14-s − 0.305·15-s + 0.250·16-s + 0.281·17-s − 0.235·18-s + 0.264·20-s − 0.116·21-s − 0.400·22-s − 1.39·23-s + 0.204·24-s − 0.719·25-s + 0.760·26-s − 0.192·27-s + 0.100·28-s − 0.746·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 1.18T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 - 2.42T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 + 0.327T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 - 5.87T + 97T^{2} \) |
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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
−8.811624079176334604802416949093, −7.78918456147384585372173394774, −7.28704779297971240863843621219, −6.26289952371780842150357259451, −5.75177842599285797101495201610, −4.76738946898824609560017633027, −3.74148417606935645520749601608, −2.36786985556159669651503031299, −1.51343651624321780601306815271, 0,
1.51343651624321780601306815271, 2.36786985556159669651503031299, 3.74148417606935645520749601608, 4.76738946898824609560017633027, 5.75177842599285797101495201610, 6.26289952371780842150357259451, 7.28704779297971240863843621219, 7.78918456147384585372173394774, 8.811624079176334604802416949093