L(s) = 1 | + (−0.891 + 0.453i)5-s + (0.142 − 0.896i)13-s + (0.734 + 1.44i)17-s + (0.587 − 0.809i)25-s + (0.0966 + 0.297i)29-s + (−0.309 − 0.0489i)37-s + (0.533 + 0.734i)41-s + i·49-s + (−0.280 + 0.550i)53-s + (1.53 + 1.11i)61-s + (0.280 + 0.863i)65-s + (1.76 − 0.278i)73-s + (−1.30 − 0.951i)85-s + (1.59 + 1.16i)89-s + (−0.896 + 1.76i)97-s + ⋯ |
L(s) = 1 | + (−0.891 + 0.453i)5-s + (0.142 − 0.896i)13-s + (0.734 + 1.44i)17-s + (0.587 − 0.809i)25-s + (0.0966 + 0.297i)29-s + (−0.309 − 0.0489i)37-s + (0.533 + 0.734i)41-s + i·49-s + (−0.280 + 0.550i)53-s + (1.53 + 1.11i)61-s + (0.280 + 0.863i)65-s + (1.76 − 0.278i)73-s + (−1.30 − 0.951i)85-s + (1.59 + 1.16i)89-s + (−0.896 + 1.76i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.038323649\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038323649\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.891 - 0.453i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.734 - 1.44i)T + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.280 - 0.550i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−8.597483381082438061579901515205, −8.021534251378567756853660791092, −7.53908006766832770996442794512, −6.59681238637636431393394734477, −5.91932988608958341518442558800, −5.04453938162711661795476470647, −4.02162037068275160900945817177, −3.45223840438058805880082510426, −2.55083874842322844418643324634, −1.12747058406160642558819075345,
0.73293158763106898779783793541, 2.08896588780027101286253778515, 3.26171681890443191568351265737, 3.98041917999723148351395345492, 4.85188153551205455069845242537, 5.42718957846115072003719631262, 6.59818709758836535033791931756, 7.19615765648904113584224126906, 7.900248868954383470748883081983, 8.600362995951573995479272816106