L(s) = 1 | + i·2-s + 3.40i·3-s − 4-s − 3.40·6-s − 0.772i·7-s − i·8-s − 8.57·9-s + 1.54·11-s − 3.40i·12-s − 3.85i·13-s + 0.772·14-s + 16-s − 1.22i·17-s − 8.57i·18-s − 6.80·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.96i·3-s − 0.5·4-s − 1.38·6-s − 0.292i·7-s − 0.353i·8-s − 2.85·9-s + 0.466·11-s − 0.982i·12-s − 1.06i·13-s + 0.206·14-s + 0.250·16-s − 0.297i·17-s − 2.02i·18-s − 1.56·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2176243931\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2176243931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 3.40iT - 3T^{2} \) |
| 7 | \( 1 + 0.772iT - 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 + 3.85iT - 13T^{2} \) |
| 17 | \( 1 + 1.22iT - 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 + 3.40iT - 23T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 0.454iT - 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 + 3.86iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 0.318iT - 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 13.6iT - 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 9.92iT - 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8iT - 83T^{2} \) |
| 89 | \( 1 - 1.71T + 89T^{2} \) |
| 97 | \( 1 - 10.9iT - 97T^{2} \) |
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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
−9.342853288062237407826096823811, −8.792854848084977833799238914301, −8.144925291437221285557315551917, −6.95964627326969769802381440585, −5.84134039674881800887001480902, −5.34145005383028637226926434298, −4.28218266764092909919266083303, −3.87590611159362690600602017296, −2.72045447453297021082354482499, −0.084140855267864550246695129046,
1.53314618546455915126020148881, 2.02894207808936407266194157926, 3.15096420029114320446075377747, 4.38980052225456695597415621333, 5.78512556656091652617852662792, 6.31886990202820317296293789613, 7.21690247751807769598771207093, 7.908049929296893064457756334304, 8.923785033516019325799403895069, 9.161688248523255459413092818748