| L(s) = 1 | + i·2-s + 4-s + (1 + 2i)5-s + 3i·8-s + (−2 + i)10-s + 6·11-s − 2i·13-s − 16-s + 4i·17-s − 6·19-s + (1 + 2i)20-s + 6i·22-s + (−3 + 4i)25-s + 2·26-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s + (0.447 + 0.894i)5-s + 1.06i·8-s + (−0.632 + 0.316i)10-s + 1.80·11-s − 0.554i·13-s − 0.250·16-s + 0.970i·17-s − 1.37·19-s + (0.223 + 0.447i)20-s + 1.27i·22-s + (−0.600 + 0.800i)25-s + 0.392·26-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533903902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533903902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.152015619096937725207148203158, −8.412928316515727147529402649335, −7.61913039043795394897084926451, −6.74289647431043218547640321919, −6.27469358068490926207003117890, −5.86039425343849187090355705307, −4.49555301422447838119620216232, −3.55116177053758679806291577369, −2.48261377113560904590093539749, −1.54357849409318562023686122849,
0.910232915939716783487750563057, 1.75625004967303957429671382069, 2.68955168677024235883511577316, 4.00056205547625356527093407031, 4.43550523097462089520048091438, 5.70687740042207680765704750497, 6.59765281391358485624757472496, 6.94374272381862301057236724196, 8.324086314088260903286211992260, 8.926832819636393698701746971037
