L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 4i·7-s + i·8-s − 9-s + i·12-s + i·13-s + 4·14-s + 16-s − 2i·17-s + i·18-s − 4·19-s + 4·21-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.51i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s + 0.277i·13-s + 1.06·14-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-s − 0.917·19-s + 0.872·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3521030005\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3521030005\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.769533074960395694331806283167, −8.293329139724881230151792418400, −7.13029841772736272071075069908, −6.29255337845808473098709614573, −5.48864365662071952858843922487, −4.68137354438211303368773936535, −3.46561640624047238436494549463, −2.41616805805092449875559807971, −1.88698152057395783200614594755, −0.12373399886804670219202394827,
1.48269543757084211639763743218, 3.31043715045951737008139054179, 3.98975436727305625103784005231, 4.72090186322783236501216952555, 5.69297987844155100951583072706, 6.46997782655535294855748051466, 7.49539115064719562021048168421, 7.77352877070571404646849925393, 8.912015836345967628129578832244, 9.520858046602122531956080282807