| L(s) = 1 | + 1.41·2-s + 2.00·4-s + (0.174 − 4.99i)5-s + 9.47i·7-s + 2.82·8-s + (0.246 − 7.06i)10-s + 20.3i·11-s + 6.10i·13-s + 13.4i·14-s + 4.00·16-s − 24.0·17-s + 16.0·19-s + (0.348 − 9.99i)20-s + 28.8i·22-s − 43.2·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + (0.0348 − 0.999i)5-s + 1.35i·7-s + 0.353·8-s + (0.0246 − 0.706i)10-s + 1.85i·11-s + 0.469i·13-s + 0.957i·14-s + 0.250·16-s − 1.41·17-s + 0.846·19-s + (0.0174 − 0.499i)20-s + 1.31i·22-s − 1.88·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0348 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0348 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.285489917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285489917\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.174 + 4.99i)T \) |
| good | 7 | \( 1 - 9.47iT - 49T^{2} \) |
| 11 | \( 1 - 20.3iT - 121T^{2} \) |
| 13 | \( 1 - 6.10iT - 169T^{2} \) |
| 17 | \( 1 + 24.0T + 289T^{2} \) |
| 19 | \( 1 - 16.0T + 361T^{2} \) |
| 23 | \( 1 + 43.2T + 529T^{2} \) |
| 29 | \( 1 + 19.6iT - 841T^{2} \) |
| 31 | \( 1 - 30.5T + 961T^{2} \) |
| 37 | \( 1 - 44.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 48.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 68.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 61.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 50.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 55.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 23.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 33.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 22.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 79.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 119. iT - 9.40e3T^{2} \) |
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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
−10.00503580920780240449366029764, −9.510099139205486237955583196959, −8.566814632023370082538209213607, −7.71371009888124936402487963313, −6.54563080014161383606499233151, −5.75723555429991766890905097293, −4.69110879616932347790120016116, −4.31102789531027970647751499539, −2.50467543521528875515086750124, −1.77088386278909752945789955098,
0.57353184407579564966358792444, 2.39374386602351503532443130969, 3.55291612113249915730193996666, 4.04490673645687985888627849028, 5.55884806430624341878510964468, 6.27748005154613793076439571769, 7.15166542886137924265654106407, 7.85406695278044815055366780215, 8.951653636613524908193471547773, 10.45208179569336939434695764535
