L(s) = 1 | + 2-s + i·3-s + 4-s + (−1.32 − 1.79i)5-s + i·6-s + 1.87i·7-s + 8-s − 9-s + (−1.32 − 1.79i)10-s − 1.51·11-s + i·12-s + 4.78·13-s + 1.87i·14-s + (1.79 − 1.32i)15-s + 16-s + 3.99·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.594 − 0.804i)5-s + 0.408i·6-s + 0.707i·7-s + 0.353·8-s − 0.333·9-s + (−0.420 − 0.568i)10-s − 0.455·11-s + 0.288i·12-s + 1.32·13-s + 0.500i·14-s + (0.464 − 0.343i)15-s + 0.250·16-s + 0.968·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.363714535\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.363714535\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (1.32 + 1.79i)T \) |
| 37 | \( 1 + (-6.03 + 0.734i)T \) |
good | 7 | \( 1 - 1.87iT - 7T^{2} \) |
| 11 | \( 1 + 1.51T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 - 3.99T + 17T^{2} \) |
| 19 | \( 1 - 4.68iT - 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 - 0.200iT - 29T^{2} \) |
| 31 | \( 1 - 5.79iT - 31T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 5.15iT - 47T^{2} \) |
| 53 | \( 1 + 2.22iT - 53T^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 4.09iT - 61T^{2} \) |
| 67 | \( 1 - 5.76iT - 67T^{2} \) |
| 71 | \( 1 + 9.17T + 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 5.95iT - 79T^{2} \) |
| 83 | \( 1 - 9.42iT - 83T^{2} \) |
| 89 | \( 1 + 5.47iT - 89T^{2} \) |
| 97 | \( 1 - 2.03T + 97T^{2} \) |
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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
−10.00452424568328181996093846748, −8.991346103286866035493574648984, −8.366821327942192216518095616870, −7.58347588154652382290536530960, −6.24785115072271315353129234872, −5.47382146431071830265648832287, −4.82719922069023151649933422892, −3.75624021285170581551425461826, −3.06340584402841054449206323565, −1.36313181518439232958754855129,
0.981964054686185985015671581216, 2.65250326030319616812519497575, 3.44618111827238633069411799440, 4.37406789716160584973215595646, 5.56790194297602268556806541431, 6.47060402639498086803506990145, 7.18355655912338833027616701279, 7.78021936085600977295293113048, 8.698024186836573367687069309153, 10.00257418810323587520841660572