L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + 0.999·8-s + (−0.5 + 0.866i)11-s + (0.499 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·22-s + (1 + 1.73i)23-s + (0.499 − 0.866i)24-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)26-s + 27-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + 0.999·8-s + (−0.5 + 0.866i)11-s + (0.499 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 0.999·22-s + (1 + 1.73i)23-s + (0.499 − 0.866i)24-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)26-s + 27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.165518644\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.165518644\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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
−8.746300465181569890677185170597, −7.993655932730052409396649925737, −7.22151741026484969351553466016, −7.08952523983129606988435928847, −5.49951932307031143968788975978, −4.82767970733724140174945465366, −3.57395742603145474214721015976, −3.00899310910461612365730960062, −1.87499124814292530563992315569, −1.30549339138968095083999445735,
0.915565441103698760847667141649, 2.48310011523602275638513868518, 3.70191053133206924462868227875, 4.22503186318174153264279351925, 5.22754437550127131603693896878, 6.05146604952542531929941469898, 6.53859165999034129900148501337, 7.67619608016697543773836867281, 8.344381114951706455960692511036, 8.773532678547593282894925144079