| L(s) = 1 | + (−0.866 + 0.5i)4-s + (0.499 − 0.866i)16-s + 2i·19-s + (1 + 1.73i)31-s + (−0.866 + 0.5i)49-s + (−1 + 1.73i)61-s + 0.999i·64-s + (−1 − 1.73i)76-s + (1.73 + i)79-s − 2i·109-s + ⋯ |
| L(s) = 1 | + (−0.866 + 0.5i)4-s + (0.499 − 0.866i)16-s + 2i·19-s + (1 + 1.73i)31-s + (−0.866 + 0.5i)49-s + (−1 + 1.73i)61-s + 0.999i·64-s + (−1 − 1.73i)76-s + (1.73 + i)79-s − 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8346347347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8346347347\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.866 - 0.5i)T^{2} \) |
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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
−9.499866868589679880848580788852, −8.581291809664136232183176171238, −8.120812377122483475434221426499, −7.32919989895104768329016313239, −6.30018649605527660117656714346, −5.43427152814621952618980069214, −4.58111168808769229002209689279, −3.75380965382476750922009634208, −2.94276145047643053495609732222, −1.41432114407216235094787806482,
0.67046582037962067550964381930, 2.19080080634334913300628981772, 3.37943693471356634025113228078, 4.50390847830914957492056276316, 4.95162640983320978379178929422, 5.98717710585087983607384892693, 6.69218565846470687622666223922, 7.73930843787554524128650484118, 8.476483284692617202736717066730, 9.328581091804656206013803900468
