L(s) = 1 | + 2i·5-s + 4·7-s − 2i·11-s + 2i·13-s + 2·17-s + 2i·19-s + 4·23-s + 25-s + 6i·29-s + 8i·35-s − 10i·37-s − 6·41-s − 6i·43-s + 8·47-s + 9·49-s + ⋯ |
L(s) = 1 | + 0.894i·5-s + 1.51·7-s − 0.603i·11-s + 0.554i·13-s + 0.485·17-s + 0.458i·19-s + 0.834·23-s + 0.200·25-s + 1.11i·29-s + 1.35i·35-s − 1.64i·37-s − 0.937·41-s − 0.914i·43-s + 1.16·47-s + 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.243204935\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.243204935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 14iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.793760956749197758015793851178, −8.509755686003725234299752697398, −7.29963258870782483710472540334, −7.12309300166512313565346938203, −5.84930303477066848876127724748, −5.23660894146425618050434861345, −4.24199522480338224179017852798, −3.32705409726734369332605831933, −2.28141504243485475984353723068, −1.22108427632985252910194514225,
0.923414068528069850584061886816, 1.81598920577219138278859064481, 3.03417653043187040154959811955, 4.41324315214423738789174059540, 4.85617985742319696959468116782, 5.48692317258635396260497518279, 6.62180298459201982165834510160, 7.64161367339961920027084258393, 8.117736686076816284090835377516, 8.782890792057012148717923048519