| L(s) = 1 | + (0.864 + 2.87i)3-s + 9.02i·7-s + (−7.50 + 4.96i)9-s − 21.8i·11-s + 21.6i·13-s − 12.1·17-s + 3.03·19-s + (−25.9 + 7.80i)21-s − 28.5·23-s + (−20.7 − 17.2i)27-s − 12.0i·29-s − 2.19·31-s + (62.8 − 18.9i)33-s − 0.839i·37-s + (−62.2 + 18.7i)39-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.957i)3-s + 1.28i·7-s + (−0.833 + 0.551i)9-s − 1.98i·11-s + 1.66i·13-s − 0.712·17-s + 0.159·19-s + (−1.23 + 0.371i)21-s − 1.24·23-s + (−0.768 − 0.639i)27-s − 0.415i·29-s − 0.0706·31-s + (1.90 − 0.573i)33-s − 0.0227i·37-s + (−1.59 + 0.480i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5285541793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5285541793\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.864 - 2.87i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 9.02iT - 49T^{2} \) |
| 11 | \( 1 + 21.8iT - 121T^{2} \) |
| 13 | \( 1 - 21.6iT - 169T^{2} \) |
| 17 | \( 1 + 12.1T + 289T^{2} \) |
| 19 | \( 1 - 3.03T + 361T^{2} \) |
| 23 | \( 1 + 28.5T + 529T^{2} \) |
| 29 | \( 1 + 12.0iT - 841T^{2} \) |
| 31 | \( 1 + 2.19T + 961T^{2} \) |
| 37 | \( 1 + 0.839iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 12.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 22.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 9.13T + 2.80e3T^{2} \) |
| 59 | \( 1 + 80.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 52.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 7.46T + 6.24e3T^{2} \) |
| 83 | \( 1 + 82.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 27.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 114. iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.834621830054469066550568891479, −9.149836312874498671251550769791, −8.642427841587302114308239102116, −8.007821640220399964595617239947, −6.37196023860877644587142231950, −5.92441172791972648757372874732, −4.91053078990135417387689457391, −3.95111461958264599288323273290, −3.00686910961054101931954882873, −2.01839286112473297553573198588,
0.14277750999915893079752745281, 1.42709930119404368959080536012, 2.48338485081820283583871739536, 3.70499734647943410965322748402, 4.65701357916209977547455274602, 5.79219297752289244303233263970, 6.86677980209037421066800014542, 7.42305261173148530764522537828, 7.908625687510151878109093861263, 8.961509526480924768837310952069
