L(s) = 1 | − 1.70i·2-s − i·3-s − 0.911·4-s − 1.70·6-s − 3.94i·7-s − 1.85i·8-s − 9-s − 5.90·11-s + 0.911i·12-s − 3.29i·13-s − 6.72·14-s − 4.99·16-s + 2.70i·17-s + 1.70i·18-s + 2.35·19-s + ⋯ |
L(s) = 1 | − 1.20i·2-s − 0.577i·3-s − 0.455·4-s − 0.696·6-s − 1.49i·7-s − 0.656i·8-s − 0.333·9-s − 1.78·11-s + 0.263i·12-s − 0.912i·13-s − 1.79·14-s − 1.24·16-s + 0.656i·17-s + 0.402i·18-s + 0.540·19-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
\(L(1)\) |
\(\approx\) |
\(1.029830013\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.029830013\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.70iT - 2T^{2} \) |
| 7 | \( 1 + 3.94iT - 7T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 3.29iT - 13T^{2} \) |
| 17 | \( 1 - 2.70iT - 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 - 0.584iT - 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 0.0208iT - 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 1.27iT - 43T^{2} \) |
| 47 | \( 1 - 5.43iT - 47T^{2} \) |
| 53 | \( 1 + 2.81iT - 53T^{2} \) |
| 59 | \( 1 - 4.69T + 59T^{2} \) |
| 61 | \( 1 - 5.58T + 61T^{2} \) |
| 67 | \( 1 - 6.03iT - 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 + 13.3iT - 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 0.781iT - 83T^{2} \) |
| 89 | \( 1 - 3.47T + 89T^{2} \) |
| 97 | \( 1 + 2.45iT - 97T^{2} \) |
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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
−8.600563103618325770652623771252, −7.72286800422401649942455735317, −7.33448980210188977763656302440, −6.30858798827037606344691283074, −5.23867754778067747269709306492, −4.24447776660971490798713210906, −3.24543832825035008740385130074, −2.58216258668697336337486227534, −1.31923020316798025819087136023, −0.37286491153632865127854876810,
2.31379087394228341437742462463, 2.88084088527466749884597029074, 4.53749526585712180250888225470, 5.27296490435453282417280577310, 5.66766871931553499956708648310, 6.60423366669396616451279753297, 7.46773837328986912189323057575, 8.293090058394282574475866305956, 8.769924575548593532033353766002, 9.580418599616956165515126863166