L(s) = 1 | + (−0.120 + 0.992i)2-s + (−0.970 − 0.239i)4-s + (0.822 − 0.568i)5-s + (0.354 − 0.935i)8-s + (−0.992 − 0.120i)9-s + (0.464 + 0.885i)10-s + (0.935 − 0.354i)13-s + (0.885 + 0.464i)16-s + (−0.424 − 0.943i)17-s + (0.239 − 0.970i)18-s + (−0.935 + 0.354i)20-s + (0.354 − 0.935i)25-s + (0.239 + 0.970i)26-s + (−1.48 − 0.180i)29-s + (−0.568 + 0.822i)32-s + ⋯ |
L(s) = 1 | + (−0.120 + 0.992i)2-s + (−0.970 − 0.239i)4-s + (0.822 − 0.568i)5-s + (0.354 − 0.935i)8-s + (−0.992 − 0.120i)9-s + (0.464 + 0.885i)10-s + (0.935 − 0.354i)13-s + (0.885 + 0.464i)16-s + (−0.424 − 0.943i)17-s + (0.239 − 0.970i)18-s + (−0.935 + 0.354i)20-s + (0.354 − 0.935i)25-s + (0.239 + 0.970i)26-s + (−1.48 − 0.180i)29-s + (−0.568 + 0.822i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.056523352\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.056523352\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.120 - 0.992i)T \) |
| 5 | \( 1 + (-0.822 + 0.568i)T \) |
| 13 | \( 1 + (-0.935 + 0.354i)T \) |
good | 3 | \( 1 + (0.992 + 0.120i)T^{2} \) |
| 7 | \( 1 + (-0.748 - 0.663i)T^{2} \) |
| 11 | \( 1 + (0.239 + 0.970i)T^{2} \) |
| 17 | \( 1 + (0.424 + 0.943i)T + (-0.663 + 0.748i)T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (1.48 + 0.180i)T + (0.970 + 0.239i)T^{2} \) |
| 31 | \( 1 + (-0.935 - 0.354i)T^{2} \) |
| 37 | \( 1 + (-0.764 + 0.527i)T + (0.354 - 0.935i)T^{2} \) |
| 41 | \( 1 + (-0.0950 + 1.57i)T + (-0.992 - 0.120i)T^{2} \) |
| 43 | \( 1 + (-0.935 + 0.354i)T^{2} \) |
| 47 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 53 | \( 1 + (0.336 + 0.748i)T + (-0.663 + 0.748i)T^{2} \) |
| 59 | \( 1 + (0.822 + 0.568i)T^{2} \) |
| 61 | \( 1 + (0.470 + 1.24i)T + (-0.748 + 0.663i)T^{2} \) |
| 67 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 71 | \( 1 + (0.992 + 0.120i)T^{2} \) |
| 73 | \( 1 + (-0.241 - 1.98i)T + (-0.970 + 0.239i)T^{2} \) |
| 79 | \( 1 + (0.885 - 0.464i)T^{2} \) |
| 83 | \( 1 + (0.120 + 0.992i)T^{2} \) |
| 89 | \( 1 + (-1.35 + 1.35i)T - iT^{2} \) |
| 97 | \( 1 + (1.45 + 0.764i)T + (0.568 + 0.822i)T^{2} \) |
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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
−8.834113778607282838562415248971, −8.082257341426857087039187809416, −7.27645928631320585384775305404, −6.37775899598746318047058305567, −5.71912147696952690802296528934, −5.33626701190515507074686341796, −4.34947943268902695048303366269, −3.37486702191908763297030554098, −2.11730142897843701721519740978, −0.68316753135625031475102886242,
1.43800749374768556670777467899, 2.28979275154842198076487084869, 3.14642691202718975582380613627, 3.90072925387215338192134542761, 4.94106406640033312155473808463, 5.89621297511083712459630030435, 6.27531514894590654400725042234, 7.52167986506771353737085281858, 8.342901111375775901035874927285, 9.003411881889357380285026308358