| L(s) = 1 | − 2-s + 4-s + 3.66i·5-s − 2.53i·7-s − 8-s − 3.66i·10-s + (−3.06 + 1.27i)11-s − 4.46i·13-s + 2.53i·14-s + 16-s − 4.49·17-s + i·19-s + 3.66i·20-s + (3.06 − 1.27i)22-s + 1.13i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.63i·5-s − 0.956i·7-s − 0.353·8-s − 1.15i·10-s + (−0.922 + 0.385i)11-s − 1.23i·13-s + 0.676i·14-s + 0.250·16-s − 1.08·17-s + 0.229i·19-s + 0.819i·20-s + (0.652 − 0.272i)22-s + 0.237i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9320646992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9320646992\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + (3.06 - 1.27i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.66iT - 5T^{2} \) |
| 7 | \( 1 + 2.53iT - 7T^{2} \) |
| 13 | \( 1 + 4.46iT - 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 23 | \( 1 - 1.13iT - 23T^{2} \) |
| 29 | \( 1 - 0.163T + 29T^{2} \) |
| 31 | \( 1 + 1.06T + 31T^{2} \) |
| 37 | \( 1 - 9.12T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 - 2.90iT - 43T^{2} \) |
| 47 | \( 1 - 1.49iT - 47T^{2} \) |
| 53 | \( 1 + 2.25iT - 53T^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 + 1.04iT - 61T^{2} \) |
| 67 | \( 1 - 3.11T + 67T^{2} \) |
| 71 | \( 1 + 8.92iT - 71T^{2} \) |
| 73 | \( 1 - 4.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.42iT - 79T^{2} \) |
| 83 | \( 1 - 8.45T + 83T^{2} \) |
| 89 | \( 1 + 15.1iT - 89T^{2} \) |
| 97 | \( 1 - 0.446T + 97T^{2} \) |
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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
−8.184586235722084199802905555849, −7.70257436419051498811015732904, −7.17986296151329307122152544221, −6.49118463059028356618715709980, −5.77714825911738872937368430421, −4.62789861133655450006098002216, −3.55726452205827922447116235399, −2.84491782738761230640128297553, −2.07935569643157136470923766197, −0.48390853253535143422705690464,
0.75455244614972764315745600954, 1.94927227210670607210340320560, 2.60976021105525618929461965427, 4.11789075193056093246131454537, 4.79491110818833358494992949056, 5.58172490338217774970303930336, 6.23891761444405934642596415698, 7.23290174915059599509545261257, 8.125465545961774192217562736646, 8.661508913746056431830344755084
