L(s) = 1 | + (−0.271 + 1.38i)2-s − 3-s + (−1.85 − 0.754i)4-s − 0.299i·5-s + (0.271 − 1.38i)6-s − 1.88·7-s + (1.55 − 2.36i)8-s + 9-s + (0.416 + 0.0814i)10-s − 2.47·11-s + (1.85 + 0.754i)12-s + 1.99i·13-s + (0.511 − 2.60i)14-s + 0.299i·15-s + (2.86 + 2.79i)16-s + 0.452·17-s + ⋯ |
L(s) = 1 | + (−0.192 + 0.981i)2-s − 0.577·3-s + (−0.926 − 0.377i)4-s − 0.134i·5-s + (0.110 − 0.566i)6-s − 0.710·7-s + (0.548 − 0.836i)8-s + 0.333·9-s + (0.131 + 0.0257i)10-s − 0.744·11-s + (0.534 + 0.217i)12-s + 0.554i·13-s + (0.136 − 0.697i)14-s + 0.0774i·15-s + (0.715 + 0.698i)16-s + 0.109·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.783759 + 0.341012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.783759 + 0.341012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.271 - 1.38i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (-2.90 + 7.65i)T \) |
good | 5 | \( 1 + 0.299iT - 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 1.99iT - 13T^{2} \) |
| 17 | \( 1 - 0.452T + 17T^{2} \) |
| 19 | \( 1 + 5.21iT - 19T^{2} \) |
| 23 | \( 1 + 0.807iT - 23T^{2} \) |
| 29 | \( 1 - 9.00T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 - 6.67iT - 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 - 2.51iT - 47T^{2} \) |
| 53 | \( 1 - 0.990iT - 53T^{2} \) |
| 59 | \( 1 - 7.56iT - 59T^{2} \) |
| 61 | \( 1 + 9.06iT - 61T^{2} \) |
| 71 | \( 1 - 0.0862iT - 71T^{2} \) |
| 73 | \( 1 - 1.91T + 73T^{2} \) |
| 79 | \( 1 - 8.30T + 79T^{2} \) |
| 83 | \( 1 - 0.827iT - 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 6.11iT - 97T^{2} \) |
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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
−10.19815618939584894482870846756, −9.468102867119053866799919557516, −8.610443753410556879657184006552, −7.71155115023407363216829347748, −6.63905845223959442734830527149, −6.31977029991516718814352079150, −5.02090054263504263254343674784, −4.50915847272661732419653250056, −2.91029007033710530096125756915, −0.75672055331039944556574291531,
0.852951875550104829337650171678, 2.53747671886258372613739419937, 3.47030329818192676542172698890, 4.65361681016110541434554051178, 5.57279042470416873372512488588, 6.59800255191303480459350451614, 7.79874913021461494490420495091, 8.529144455603966060671568390228, 9.684014187494583488581721738656, 10.32943373538909722333585428206