| L(s) = 1 | + 3-s + 2i·5-s + 4i·7-s + 9-s − 6i·11-s + (3 − 2i)13-s + 2i·15-s − 2·17-s + 4i·21-s − 8·23-s + 25-s + 27-s + 2·29-s − 8i·31-s − 6i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894i·5-s + 1.51i·7-s + 0.333·9-s − 1.80i·11-s + (0.832 − 0.554i)13-s + 0.516i·15-s − 0.485·17-s + 0.872i·21-s − 1.66·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s − 1.43i·31-s − 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27161 + 0.385013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27161 + 0.385013i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−13.26657220693466661731948892325, −11.88375396217184662205761068204, −11.12128195663621032232420364802, −9.984272626526803271025881911865, −8.634512573800313818344453651741, −8.218875798418116076630631770765, −6.43439919378819890018806124998, −5.67311703943898644966786049773, −3.54941629286267712884198762723, −2.51497422957877804194212538733,
1.65252021016815052804364632200, 3.96615499698305412496572344175, 4.69084294487850614818309091038, 6.69800751741339245219009105952, 7.61638043898165400830288508122, 8.733357697731066924305144083051, 9.793052053774398617031321730766, 10.60968435213668987297176670796, 12.08276686225715194715834263929, 12.94932386452613100433413645942
