| L(s) = 1 | − 1.73i·5-s + (2 + 1.73i)7-s + 3.46i·11-s − 3.46i·13-s + 5.19i·17-s − 2·19-s + 6.92i·23-s + 2.00·25-s + 6·29-s − 10·31-s + (2.99 − 3.46i)35-s − 11·37-s + 1.73i·41-s + 5.19i·43-s + 9·47-s + ⋯ |
| L(s) = 1 | − 0.774i·5-s + (0.755 + 0.654i)7-s + 1.04i·11-s − 0.960i·13-s + 1.26i·17-s − 0.458·19-s + 1.44i·23-s + 0.400·25-s + 1.11·29-s − 1.79·31-s + (0.507 − 0.585i)35-s − 1.80·37-s + 0.270i·41-s + 0.792i·43-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.569335425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569335425\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 1.73iT - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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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
−8.872622283220259127185845127535, −8.131851315225435693770630618465, −7.62112508302919210895397950862, −6.61181337334973260989875117741, −5.57312619810038405231423009757, −5.15556415856095759135825336559, −4.32635082558230335095869552433, −3.34211146395643793259553752300, −2.05829192424773217696333797947, −1.34982146150484582549255033383,
0.49525080591955792677957552443, 1.88721971139079786726223374655, 2.88612970688943832530762588226, 3.78689889525344726823307404797, 4.65362491375754674485683089362, 5.42207609828340050113608614520, 6.58768498218918007471176386339, 6.92112389974738081050197346690, 7.72338518505025587316238289617, 8.684465308900046124477111975093
