| L(s) = 1 | + i·4-s + (1 + i)7-s − 16-s + (1 − i)19-s + i·25-s + (−1 + i)28-s + (−1 + i)31-s + (−1 − i)37-s + i·49-s − i·64-s + (1 − i)67-s + (1 + i)73-s + (1 + i)76-s + (−1 + i)97-s − 100-s + ⋯ |
| L(s) = 1 | + i·4-s + (1 + i)7-s − 16-s + (1 − i)19-s + i·25-s + (−1 + i)28-s + (−1 + i)31-s + (−1 − i)37-s + i·49-s − i·64-s + (1 − i)67-s + (1 + i)73-s + (1 + i)76-s + (−1 + i)97-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201909156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201909156\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - iT^{2} \) |
| 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (-1 - i)T + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1 - i)T - iT^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−9.492389179499621192347036867256, −8.936301586420276716704119228963, −8.292715549792970586562583242963, −7.46548761150637029561010310272, −6.84212553812636642622348832502, −5.44778760234932995012295545444, −4.98999663676339519289227836343, −3.77475559309054979235419571932, −2.85421897115998887987872250432, −1.81695550304147326141005723360,
1.07304974082192346023987081374, 2.05742189762647548783545866750, 3.64893983664295827796460757339, 4.58013366071585867010715589083, 5.31575483310630235484014456841, 6.17205468239825203767644146189, 7.13800899756430339627551879203, 7.84138029764644006451824924470, 8.680573982591645169164372269874, 9.771287775595117019428877122743
