L(s) = 1 | − 2.35i·2-s − i·3-s − 3.52·4-s − 2.35·6-s − 3.48i·7-s + 3.58i·8-s − 9-s + 2.93·11-s + 3.52i·12-s − 1.87i·13-s − 8.18·14-s + 1.38·16-s − 6.78i·17-s + 2.35i·18-s + 2.94·19-s + ⋯ |
L(s) = 1 | − 1.66i·2-s − 0.577i·3-s − 1.76·4-s − 0.959·6-s − 1.31i·7-s + 1.26i·8-s − 0.333·9-s + 0.883·11-s + 1.01i·12-s − 0.520i·13-s − 2.18·14-s + 0.345·16-s − 1.64i·17-s + 0.554i·18-s + 0.676·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.387568761\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387568761\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.35iT - 2T^{2} \) |
| 7 | \( 1 + 3.48iT - 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.87iT - 13T^{2} \) |
| 17 | \( 1 + 6.78iT - 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 + 5.49iT - 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 - 0.418T + 31T^{2} \) |
| 37 | \( 1 + 5.23iT - 37T^{2} \) |
| 41 | \( 1 - 1.67T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 7.49iT - 47T^{2} \) |
| 53 | \( 1 - 3.70iT - 53T^{2} \) |
| 59 | \( 1 - 7.10T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 0.728T + 71T^{2} \) |
| 73 | \( 1 + 3.59iT - 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 0.287T + 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−9.058302935508946397779575767542, −7.85775129761214966413789954773, −7.21041794743159519656291139871, −6.32973107331353178051512392192, −4.97181334414687725874439145113, −4.22580691782514172185273960244, −3.33101603411127659319011018238, −2.53098705534961033150316985573, −1.22951357290615196163277267802, −0.57807887973933184622363496553,
1.89256749181070762889954472761, 3.49821182585391797262680654246, 4.30821141409471308587377285498, 5.40768283080380680877516764788, 5.75832344750539465595124176421, 6.55412641020028438447572335800, 7.36568648387603289243279737562, 8.408735596333760692838364775342, 8.773846293104457161919988397572, 9.412814721342810807090832148538