L(s) = 1 | − 2.17i·2-s − i·3-s − 2.70·4-s + (2.17 − 0.539i)5-s − 2.17·6-s − 3.70i·7-s + 1.53i·8-s − 9-s + (−1.17 − 4.70i)10-s + 2.70i·12-s − 1.70i·13-s − 8.04·14-s + (−0.539 − 2.17i)15-s − 2.07·16-s + 6.04i·17-s + 2.17i·18-s + ⋯ |
L(s) = 1 | − 1.53i·2-s − 0.577i·3-s − 1.35·4-s + (0.970 − 0.241i)5-s − 0.885·6-s − 1.40i·7-s + 0.544i·8-s − 0.333·9-s + (−0.370 − 1.48i)10-s + 0.782i·12-s − 0.474i·13-s − 2.15·14-s + (−0.139 − 0.560i)15-s − 0.519·16-s + 1.46i·17-s + 0.511i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.538338341\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.538338341\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.17 + 0.539i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.17iT - 2T^{2} \) |
| 7 | \( 1 + 3.70iT - 7T^{2} \) |
| 13 | \( 1 + 1.70iT - 13T^{2} \) |
| 17 | \( 1 - 6.04iT - 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 + 6.34T + 31T^{2} \) |
| 37 | \( 1 + 3.41iT - 37T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 + 3.12iT - 43T^{2} \) |
| 47 | \( 1 - 2.73iT - 47T^{2} \) |
| 53 | \( 1 + 13.7iT - 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 1.84iT - 67T^{2} \) |
| 71 | \( 1 + 7.23T + 71T^{2} \) |
| 73 | \( 1 - 6.38iT - 73T^{2} \) |
| 79 | \( 1 + 7.44T + 79T^{2} \) |
| 83 | \( 1 + 7.86iT - 83T^{2} \) |
| 89 | \( 1 - 5.02T + 89T^{2} \) |
| 97 | \( 1 - 16.9iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.795687741286339520107625695798, −8.249079245522034170110419524990, −6.98854912936429585786626390129, −6.40897800630776521144153060673, −5.25513931032329827894036158090, −4.21710741495052029716873067932, −3.48376953884942408299214836753, −2.28907517180878215660442549045, −1.54850400066116099451268520209, −0.55269930545472202759070408657,
2.08420876298784664573500360126, 3.03368873417846452954143558752, 4.58962581006686642320657262286, 5.29062282498565443048582446566, 5.78118010404592657321429232920, 6.56903430817682529018972536222, 7.23704445112709657844521941884, 8.384478161216312750746290313989, 8.978638358066085212056747809362, 9.413322252130390890894880186050