| L(s) = 1 | + i·2-s − 4-s + 0.366·5-s + (−1.91 + 1.82i)7-s − i·8-s + 0.366i·10-s + 0.669i·11-s + 1.00i·13-s + (−1.82 − 1.91i)14-s + 16-s − 4.98·17-s − 6.35i·19-s − 0.366·20-s − 0.669·22-s + 7.69i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.163·5-s + (−0.723 + 0.690i)7-s − 0.353i·8-s + 0.115i·10-s + 0.201i·11-s + 0.277i·13-s + (−0.488 − 0.511i)14-s + 0.250·16-s − 1.21·17-s − 1.45i·19-s − 0.0819·20-s − 0.142·22-s + 1.60i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2470381592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2470381592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.91 - 1.82i)T \) |
| good | 5 | \( 1 - 0.366T + 5T^{2} \) |
| 11 | \( 1 - 0.669iT - 11T^{2} \) |
| 13 | \( 1 - 1.00iT - 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 + 6.35iT - 19T^{2} \) |
| 23 | \( 1 - 7.69iT - 23T^{2} \) |
| 29 | \( 1 - 1.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 + 4.95iT - 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.49iT - 71T^{2} \) |
| 73 | \( 1 - 4.07iT - 73T^{2} \) |
| 79 | \( 1 - 8.35T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 17.2iT - 97T^{2} \) |
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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
−9.994676626594664641850231978518, −9.291933354076162310762617408144, −8.869900575165441251065129027101, −7.74337632612279198102975910544, −6.90662134000255742431029068770, −6.22030694578995185818246179864, −5.36780044545850823866469890403, −4.43363100096529758111346850037, −3.27005753359522848360298355153, −2.03415686097507298463707082038,
0.10232005217653415599073482628, 1.70107962364628593388021283948, 2.95527737299856237310133135841, 3.87610715892483598513219070461, 4.72315899203066941165651730530, 5.98996639116732104777124744367, 6.67276168484718699853416574843, 7.80317960859178778564663867657, 8.629170296558751764891893620082, 9.461670658531800498403458204930
