L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (1 − 1.73i)19-s + (0.5 + 0.866i)25-s + 0.999·27-s + (−1 − 1.73i)31-s + (1 − 1.73i)37-s − 1.99·57-s + (0.499 − 0.866i)75-s + (−0.5 − 0.866i)81-s + (−0.999 + 1.73i)93-s + (−1 + 1.73i)103-s + (−1 − 1.73i)109-s − 1.99·111-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (1 − 1.73i)19-s + (0.5 + 0.866i)25-s + 0.999·27-s + (−1 − 1.73i)31-s + (1 − 1.73i)37-s − 1.99·57-s + (0.499 − 0.866i)75-s + (−0.5 − 0.866i)81-s + (−0.999 + 1.73i)93-s + (−1 + 1.73i)103-s + (−1 − 1.73i)109-s − 1.99·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9358111307\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9358111307\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.095985056644382999589452427633, −7.988972643881916246526790481354, −7.35045076792803092660656494565, −6.82814991168123800432288051962, −5.79817892981730959401129941485, −5.27036874650577185583486301279, −4.25571777076853829776551171063, −2.98800308743506057526001928020, −2.08315057583821279625076795836, −0.75155836789828295863544581329,
1.34497072428732213919284917472, 2.96026680797038438561041820804, 3.69852074162922439106140613861, 4.63949166347080864425977324347, 5.37530197767016374525331139885, 6.11391631326066834002344188924, 6.90610897171116143590507148111, 7.958411572793768958225092059465, 8.643294438049273805902338103224, 9.517379058209313772826750562947