| L(s) = 1 | − i·3-s + 4i·7-s − 9-s + 2i·13-s + 6i·17-s + 4·19-s + 4·21-s + i·27-s − 6·29-s − 8·31-s − 2i·37-s + 2·39-s − 6·41-s + 4i·43-s − 9·49-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s + 0.872·21-s + 0.192i·27-s − 1.11·29-s − 1.43·31-s − 0.328i·37-s + 0.320·39-s − 0.937·41-s + 0.609i·43-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7495184164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7495184164\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.537395137449503963203691087052, −7.966263376202387062752261235190, −7.11416967298402658014540010980, −6.38421865062114281646091783512, −5.63252409518542391525213956678, −5.25694533812978407052290526532, −3.98344400497615501100868244564, −3.17936051963050208815382389477, −2.12628002976967149965764387077, −1.59155952422769810999849494925,
0.20269585753409633530984875096, 1.26626555704771791504293917164, 2.68190670110707957504831908173, 3.58157274485400946865577139421, 4.06876884228442224896906616215, 5.13574751213283209023778066655, 5.46589143034714443083993940067, 6.75285843438458058155856004519, 7.29191628020203835954349274503, 7.80235107634565824670375211405
