| L(s) = 1 | + i·2-s − 4-s − 0.929·5-s + (−2.07 + 1.63i)7-s − i·8-s − 0.929i·10-s − i·11-s + (−1.63 − 2.07i)14-s + 16-s + 2.34·17-s − 6.87i·19-s + 0.929·20-s + 22-s − 3.04i·23-s − 4.13·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.415·5-s + (−0.785 + 0.619i)7-s − 0.353i·8-s − 0.294i·10-s − 0.301i·11-s + (−0.437 − 0.555i)14-s + 0.250·16-s + 0.569·17-s − 1.57i·19-s + 0.207·20-s + 0.213·22-s − 0.635i·23-s − 0.827·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9911541602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9911541602\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.07 - 1.63i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 0.929T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 + 6.87iT - 19T^{2} \) |
| 23 | \( 1 + 3.04iT - 23T^{2} \) |
| 29 | \( 1 + 6.39iT - 29T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 1.85T + 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 + 0.0878T + 67T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 12.0iT - 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 4.06T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 15.6iT - 97T^{2} \) |
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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
−9.223868130625018907493229407161, −8.821151083270053985002468586618, −7.84689976361186353156193807878, −7.07724719040026496714932520636, −6.27358921555073563231201770339, −5.53157706482099015139187457267, −4.55567921415913462738454157021, −3.52204238362193075613537981422, −2.51482080409563649379995777178, −0.48279924590825292288324225344,
1.09413389785414146625048930157, 2.48072636785076359739177355217, 3.77256139920138794970030946774, 3.98706438061649028334489964502, 5.43998830676597376055175431950, 6.20481543679167642135611798286, 7.52079789816293653756120192809, 7.79305893743532675439174257360, 9.116107482911826794140995892761, 9.674213966643654799726034791029
