| L(s) = 1 | + (0.5 + 1.32i)2-s + (−1.50 + 1.32i)4-s − 1.73i·5-s + (−2.50 − 1.32i)8-s + (2.29 − 0.866i)10-s − 2.64i·11-s + 3.46i·13-s + (0.500 − 3.96i)16-s + 6.92i·17-s + (2.29 + 2.59i)20-s + (3.50 − 1.32i)22-s − 5.29i·23-s + 2.00·25-s + (−4.58 + 1.73i)26-s + 5·29-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.935i)2-s + (−0.750 + 0.661i)4-s − 0.774i·5-s + (−0.883 − 0.467i)8-s + (0.724 − 0.273i)10-s − 0.797i·11-s + 0.960i·13-s + (0.125 − 0.992i)16-s + 1.68i·17-s + (0.512 + 0.580i)20-s + (0.746 − 0.282i)22-s − 1.10i·23-s + 0.400·25-s + (−0.898 + 0.339i)26-s + 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.803920905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.803920905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.5 - 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 - 9.16T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 7.93iT - 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−9.114133488726863122686315851967, −8.432209991816665304972612842773, −8.183667990817238288232031782389, −6.87751428309633074833202804039, −6.29274147549980074972323617324, −5.53351863440037049129406782224, −4.49713806362806097490748281410, −4.03818041839663575248585315969, −2.73675951467149930762715460838, −1.03098934758103407185029868349,
0.78024984315688695184200424687, 2.32588970078505830074198804204, 2.96307984160516419896287361220, 3.91846979257886618264801626085, 4.99757366672446933341466447486, 5.57453081805246188154603541084, 6.79310375428092304739491733911, 7.40996283774560999288913840820, 8.521441846844305282411369856319, 9.379396173313334218223433325631
