| L(s) = 1 | + 3.46i·5-s − i·7-s − 17.3·11-s − 1.73i·13-s + 6·17-s − 1.73·19-s + 30i·23-s + 13.0·25-s − 20.7i·29-s − 14i·31-s + 3.46·35-s − 19.0i·37-s − 48·41-s − 20.7·43-s − 66i·47-s + ⋯ |
| L(s) = 1 | + 0.692i·5-s − 0.142i·7-s − 1.57·11-s − 0.133i·13-s + 0.352·17-s − 0.0911·19-s + 1.30i·23-s + 0.520·25-s − 0.716i·29-s − 0.451i·31-s + 0.0989·35-s − 0.514i·37-s − 1.17·41-s − 0.483·43-s − 1.40i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.066326589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066326589\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46iT - 25T^{2} \) |
| 7 | \( 1 + iT - 49T^{2} \) |
| 11 | \( 1 + 17.3T + 121T^{2} \) |
| 13 | \( 1 + 1.73iT - 169T^{2} \) |
| 17 | \( 1 - 6T + 289T^{2} \) |
| 19 | \( 1 + 1.73T + 361T^{2} \) |
| 23 | \( 1 - 30iT - 529T^{2} \) |
| 29 | \( 1 + 20.7iT - 841T^{2} \) |
| 31 | \( 1 + 14iT - 961T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 48T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 66iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 48.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49T + 5.32e3T^{2} \) |
| 79 | \( 1 + 83iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 66T + 7.92e3T^{2} \) |
| 97 | \( 1 - 107T + 9.40e3T^{2} \) |
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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
−8.939011234984704613093867605737, −8.008741218301624768643820019817, −7.46775547395416317548712219925, −6.66712550006031005924865317751, −5.62044634308514273334210133300, −5.04155306834375656935811507429, −3.75621958791742054510380953727, −2.94364649181144867130936339030, −1.98175002532849832403377901390, −0.31830479102582335609869801709,
1.00274658529169511757668135364, 2.35069469468591793918603266635, 3.24209689520940566963425092489, 4.60916149239987133887358081849, 5.07317408768245201954419326770, 5.97935966202876790794441695732, 6.97318509048798846512524995853, 7.85305502183637492346902927516, 8.534807103675164021749422337273, 9.118295793356615432990396511590
