L(s) = 1 | + (0.831 + 0.831i)2-s + (−0.144 + 0.0600i)3-s + 0.381i·4-s + (−0.170 − 0.0705i)6-s + (0.744 − 1.79i)7-s + (0.513 − 0.513i)8-s + (−0.689 + 0.689i)9-s + (−0.0229 − 0.0553i)12-s + (2.11 − 0.874i)14-s + 1.23·16-s + (−0.309 + 0.951i)17-s − 1.14·18-s + 0.305i·21-s + (−0.0436 + 0.105i)24-s + (−0.707 + 0.707i)25-s + ⋯ |
L(s) = 1 | + (0.831 + 0.831i)2-s + (−0.144 + 0.0600i)3-s + 0.381i·4-s + (−0.170 − 0.0705i)6-s + (0.744 − 1.79i)7-s + (0.513 − 0.513i)8-s + (−0.689 + 0.689i)9-s + (−0.0229 − 0.0553i)12-s + (2.11 − 0.874i)14-s + 1.23·16-s + (−0.309 + 0.951i)17-s − 1.14·18-s + 0.305i·21-s + (−0.0436 + 0.105i)24-s + (−0.707 + 0.707i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.467690454\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.467690454\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (0.309 - 0.951i)T \) |
| 47 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.831 - 0.831i)T + iT^{2} \) |
| 3 | \( 1 + (0.144 - 0.0600i)T + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 7 | \( 1 + (-0.744 + 1.79i)T + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (1.84 - 0.763i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 59 | \( 1 + (1.14 - 1.14i)T - iT^{2} \) |
| 61 | \( 1 + (-0.652 + 1.57i)T + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.20 + 0.497i)T + (0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (-1.20 - 0.497i)T + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 + 1.17iT - T^{2} \) |
| 97 | \( 1 + (0.399 + 0.965i)T + (-0.707 + 0.707i)T^{2} \) |
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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
−10.69560395836249096797765019838, −9.932702572448796884036303536248, −8.456788595165184040628255638939, −7.68358171896417580852955392463, −7.06156797750850884516471775523, −6.08268329481494116685183119599, −5.14095174726300845463300754406, −4.40389014307780480914153795140, −3.58089174005224129702127668263, −1.55137492389192093949885263424,
2.03377961245795082116856778826, 2.76180661304722172829474014509, 3.88564385904656786407428291302, 5.22770284877695370330538520179, 5.48014779234737893977896054464, 6.74048579145474274465647678368, 8.136015347281656707824731521041, 8.707786781304350002030698189121, 9.579304294547858737368971953722, 10.83878884212750924866212340532