| L(s) = 1 | − 11.5i·3-s − 27.0i·7-s + 110.·9-s + 226.·11-s + 511. i·13-s − 387. i·17-s + 1.33e3·19-s − 311.·21-s + 545. i·23-s − 4.07e3i·27-s + 4.63e3·29-s + 2.99e3·31-s − 2.60e3i·33-s + 1.26e3i·37-s + 5.89e3·39-s + ⋯ |
| L(s) = 1 | − 0.739i·3-s − 0.208i·7-s + 0.453·9-s + 0.563·11-s + 0.839i·13-s − 0.325i·17-s + 0.848·19-s − 0.154·21-s + 0.214i·23-s − 1.07i·27-s + 1.02·29-s + 0.559·31-s − 0.416i·33-s + 0.151i·37-s + 0.620·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.184732766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184732766\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 11.5iT - 243T^{2} \) |
| 7 | \( 1 + 27.0iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 226.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 511. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 387. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 545. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.30e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.89e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.93e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 9.06e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.55e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.01e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.32e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 4.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.05e4iT - 8.58e9T^{2} \) |
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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
−11.77794282143940038979626971167, −10.37701801369147023043662322307, −9.431678667350035711509418431623, −8.286811720222808775627123115833, −7.12470316782803923429317387202, −6.53995649664616726751603688828, −4.99296988038484382731376090583, −3.68841743811035441554087435286, −2.01029974440302141693590370904, −0.826974206065066676624359460516,
1.15156384664476019735896828331, 2.98920402081083469267602018400, 4.19879431243808808820764684618, 5.27122104770866845200703011982, 6.52789886865690339368444716023, 7.78168189526052989049280944731, 8.911478595007957091740517695984, 9.889327075000933717553596911233, 10.57007493975084953238511686564, 11.72969099020910100232826714806
