| L(s) = 1 | + 2-s + 4-s − 5-s − 3.75i·7-s + 8-s − 10-s − 0.476·11-s + 4.74i·13-s − 3.75i·14-s + 16-s + 4.08i·17-s − 3.12·19-s − 20-s − 0.476·22-s − 0.982i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.42i·7-s + 0.353·8-s − 0.316·10-s − 0.143·11-s + 1.31i·13-s − 1.00i·14-s + 0.250·16-s + 0.990i·17-s − 0.717·19-s − 0.223·20-s − 0.101·22-s − 0.204i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.332365266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332365266\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (7.79 + 2.51i)T \) |
| good | 7 | \( 1 + 3.75iT - 7T^{2} \) |
| 11 | \( 1 + 0.476T + 11T^{2} \) |
| 13 | \( 1 - 4.74iT - 13T^{2} \) |
| 17 | \( 1 - 4.08iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 0.982iT - 23T^{2} \) |
| 29 | \( 1 - 5.08iT - 29T^{2} \) |
| 31 | \( 1 - 1.51iT - 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 1.30iT - 43T^{2} \) |
| 47 | \( 1 + 0.0921iT - 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 11.8iT - 59T^{2} \) |
| 61 | \( 1 + 12.8iT - 61T^{2} \) |
| 71 | \( 1 - 13.0iT - 71T^{2} \) |
| 73 | \( 1 + 7.77T + 73T^{2} \) |
| 79 | \( 1 - 9.74iT - 79T^{2} \) |
| 83 | \( 1 - 15.7iT - 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 6.02iT - 97T^{2} \) |
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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
−8.200269078314600691592253572772, −7.38432267658939088851949475488, −6.72387652785032121298430078384, −6.44338496377243964970509789915, −5.19118544718828195030661879438, −4.56320745598989708065292742955, −3.85471588365932894803396158923, −3.46924858132538720774385618263, −2.12436144780281080625246270365, −1.26772222788650241996225724607,
0.25010096201736592292993700852, 1.83619809114205224329121672555, 2.78613202759343592112075781216, 3.19470519857481493864720149472, 4.32564461652904398338864338516, 5.02629617628346205039839719276, 5.72607973928192052940418070964, 6.13883544927354333860459535902, 7.23005319241468757565077442925, 7.72291837551929781184201289613
