L(s) = 1 | − i·2-s − 4-s + (0.707 + 0.707i)5-s − i·7-s + i·8-s + (0.707 − 0.707i)10-s + 1.41i·13-s − 14-s + 16-s + 1.41·19-s + (−0.707 − 0.707i)20-s + 1.00i·25-s + 1.41·26-s + i·28-s − i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (0.707 + 0.707i)5-s − i·7-s + i·8-s + (0.707 − 0.707i)10-s + 1.41i·13-s − 14-s + 16-s + 1.41·19-s + (−0.707 − 0.707i)20-s + 1.00i·25-s + 1.41·26-s + i·28-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.260510600\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260510600\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.384924246724247000453699452686, −8.431327821139482313600047996792, −7.37551162160778305985684259661, −6.82763777237242469461493147675, −5.78306524062205899355110329751, −4.88341680713986146242717727265, −3.98230081103615518005565762796, −3.24714793227316224933648503366, −2.20431174424225376329667952216, −1.25481888847999473617822282080,
1.04773547050321697309876268420, 2.57122152479373679057078250940, 3.63773937410631060655005934609, 4.93841222341856648093892857258, 5.41834087466413072585229649494, 5.87576438843376658651949441175, 6.82309158409667021732299080566, 7.81537087955322464454039688412, 8.370683344412948174289370838900, 9.042427766819186470553557295708