| L(s) = 1 | − 3-s + i·5-s + 4.70i·7-s + 9-s − 2.70i·11-s + (2 − 3i)13-s − i·15-s + 6.70·17-s − 7.40i·19-s − 4.70i·21-s + 2.70·23-s − 25-s − 27-s + 2·29-s + 9.40i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447i·5-s + 1.77i·7-s + 0.333·9-s − 0.814i·11-s + (0.554 − 0.832i)13-s − 0.258i·15-s + 1.62·17-s − 1.69i·19-s − 1.02i·21-s + 0.563·23-s − 0.200·25-s − 0.192·27-s + 0.371·29-s + 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.467485232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.467485232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-2 + 3i)T \) |
| good | 7 | \( 1 - 4.70iT - 7T^{2} \) |
| 11 | \( 1 + 2.70iT - 11T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 7.40iT - 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 9.40iT - 31T^{2} \) |
| 37 | \( 1 - 2.70iT - 37T^{2} \) |
| 41 | \( 1 - 8.70iT - 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 - 0.596iT - 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 0.701iT - 71T^{2} \) |
| 73 | \( 1 + 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 - 9.40iT - 83T^{2} \) |
| 89 | \( 1 - 4.70iT - 89T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
| 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
−9.538298572865960518248290423093, −8.706376801689180841335042376516, −8.144998027981335752862421355343, −6.99283694317709395987181788345, −6.14409439771898204304702971522, −5.52434472409603220782377137220, −4.91388579475347018628894173506, −3.20602186740171614901742670909, −2.79417707023765875861947110796, −1.10347828958971854439934056747,
0.795757558177310728955073776668, 1.74885985166863053704684536533, 3.79222527395488031424655847036, 4.04073763065268806331557545790, 5.17602536360874176620265530429, 6.02346402499296233392244566944, 7.05675592422637277268834852976, 7.53284828160321113745482430398, 8.370259517727605205008542408426, 9.650985598829725419863561454865
