| L(s) = 1 | − 2i·2-s − 4·4-s + 8i·8-s − 10i·13-s + 16·16-s − 16i·17-s − 20·26-s − 40·29-s − 32i·32-s − 32·34-s + 70i·37-s − 80·41-s − 49·49-s + 40i·52-s − 56i·53-s + ⋯ |
| L(s) = 1 | − i·2-s − 4-s + i·8-s − 0.769i·13-s + 16-s − 0.941i·17-s − 0.769·26-s − 1.37·29-s − i·32-s − 0.941·34-s + 1.89i·37-s − 1.95·41-s − 0.999·49-s + 0.769i·52-s − 1.05i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2206651424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2206651424\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 + 16iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 + 40T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 70iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 80T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 56iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 110iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 160T + 7.92e3T^{2} \) |
| 97 | \( 1 - 130iT - 9.40e3T^{2} \) |
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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
−9.652197036845686763356614594240, −8.612790887128455351314637716683, −7.927574606019179938259707335583, −6.80110916338318980829006161127, −5.50299938187821274104080290517, −4.82791221251137119207238128926, −3.61344297103619671108836330955, −2.78356309920885525975133623691, −1.48799123756900941233887146069, −0.07288480697499949785533325705,
1.72606031449390733016546906619, 3.52004137090482419973307972892, 4.36107934257471320419583540634, 5.41886994382864931517914719396, 6.22014969446764431483932086254, 7.07611940513120865775326939646, 7.85039533485365692458881010196, 8.763785647629164018019312892561, 9.376489537846994906992010782212, 10.29728063752796350354402982503
