| L(s) = 1 | + 2.23i·5-s + 11.7i·7-s + 9.80i·11-s + 0.546·13-s − 3.84i·17-s − 12.5i·19-s + (−22.2 − 5.78i)23-s − 5.00·25-s − 37.7·29-s − 17.3·31-s − 26.2·35-s + 65.6i·37-s + 56.5·41-s − 67.9i·43-s − 43.0·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.67i·7-s + 0.891i·11-s + 0.0420·13-s − 0.226i·17-s − 0.661i·19-s + (−0.967 − 0.251i)23-s − 0.200·25-s − 1.30·29-s − 0.560·31-s − 0.748·35-s + 1.77i·37-s + 1.38·41-s − 1.58i·43-s − 0.915·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01585285453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01585285453\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (22.2 + 5.78i)T \) |
| good | 7 | \( 1 - 11.7iT - 49T^{2} \) |
| 11 | \( 1 - 9.80iT - 121T^{2} \) |
| 13 | \( 1 - 0.546T + 169T^{2} \) |
| 17 | \( 1 + 3.84iT - 289T^{2} \) |
| 19 | \( 1 + 12.5iT - 361T^{2} \) |
| 29 | \( 1 + 37.7T + 841T^{2} \) |
| 31 | \( 1 + 17.3T + 961T^{2} \) |
| 37 | \( 1 - 65.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 56.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 67.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 43.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 53.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 24.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 19.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 71.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 44.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 90.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 30.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 61.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 123. iT - 9.40e3T^{2} \) |
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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
−8.780642830713563886362433900460, −8.100259316640750966102370294525, −7.26071363196614643071908473943, −6.56129097734674834228946390693, −5.76930379803424480951155075980, −5.18793990609989558196456461562, −4.28668888050699680927987681392, −3.23222666925121127216149332676, −2.38032512782894753298288264431, −1.80283822374920759472085284430,
0.00352350933091319713468540860, 0.949166592928053364047931939970, 1.84552342171713758779349009388, 3.26914204058296696721778805333, 3.96091105542365674441641644086, 4.46417298641926090427865145327, 5.69601138643766167190403373613, 6.08214963110917127197400925649, 7.24640425014755021924596004351, 7.65927534811813488886884872984
