| L(s) = 1 | − i·3-s + i·7-s − 9-s + i·13-s + 3·19-s + 21-s − 4i·23-s + i·27-s − 4·29-s + 7·31-s + 6i·37-s + 39-s + 6·41-s − 9i·43-s + 6i·47-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.277i·13-s + 0.688·19-s + 0.218·21-s − 0.834i·23-s + 0.192i·27-s − 0.742·29-s + 1.25·31-s + 0.986i·37-s + 0.160·39-s + 0.937·41-s − 1.37i·43-s + 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.755419582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.755419582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 9iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 3iT - 97T^{2} \) |
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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
−8.811838455955098507971772836951, −8.146703323048170511522494536260, −7.35469172488665599291926687619, −6.61455676196099202417718681674, −5.87021932336371282003594363918, −5.04570413790200195647156031484, −4.06268406448397177082965768813, −2.94365235085101438186982523923, −2.10097131688207670852488339887, −0.836644953861283458812990788895,
0.885222922331044474368563886673, 2.35204244097262856715214410224, 3.42247850540730106229925564516, 4.11568056353240068866076051385, 5.10138005572477450770437735022, 5.74680830961510360846998976860, 6.72021627233839123187251155989, 7.59104091284978371235085103925, 8.198711330571139305621037745169, 9.216231332387551407410289018942
