L(s) = 1 | − 0.792·3-s + 1.37i·5-s − 2.37·9-s + 3.31·11-s − 1.08i·15-s − 6.13i·23-s + 3.11·25-s + 4.25·27-s + 9.30i·31-s − 2.62·33-s − 12.1i·37-s − 3.25i·45-s + 6.63i·47-s − 7·49-s + 6i·53-s + ⋯ |
L(s) = 1 | − 0.457·3-s + 0.613i·5-s − 0.790·9-s + 1.00·11-s − 0.280i·15-s − 1.27i·23-s + 0.623·25-s + 0.819·27-s + 1.67i·31-s − 0.457·33-s − 1.99i·37-s − 0.485i·45-s + 0.967i·47-s − 49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.400244177\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.400244177\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - 3.31T \) |
good | 3 | \( 1 + 0.792T + 3T^{2} \) |
| 5 | \( 1 - 1.37iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6.13iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 9.30iT - 31T^{2} \) |
| 37 | \( 1 + 12.1iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 18.8T + 89T^{2} \) |
| 97 | \( 1 + 0.116T + 97T^{2} \) |
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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
−8.797197756893491464004661276211, −8.305258316646955670496364117864, −7.07313396471423018290912941134, −6.68006841546758664560428826697, −5.90290342465244929586633228524, −5.09872904731786030269865981257, −4.13808771915349243572940621168, −3.20407719870905669622596079278, −2.33043100193886502724764979636, −0.894281837371564079975945463510,
0.64636979766129279017542263407, 1.77873190063890394447007431862, 3.08529496149198917898974371222, 3.98175261323104997263332847861, 4.93596464973607009654474574136, 5.56839526604721845274820224436, 6.37422738997817128018644918393, 7.04370158200322481908005529764, 8.191516766383571642296175883173, 8.571098794668130460041748243125