L(s) = 1 | + 2-s + i·3-s + 4-s + (−2 − i)5-s + i·6-s − 3i·7-s + 8-s − 9-s + (−2 − i)10-s − 3·11-s + i·12-s + 4·13-s − 3i·14-s + (1 − 2i)15-s + 16-s − 7·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408i·6-s − 1.13i·7-s + 0.353·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.904·11-s + 0.288i·12-s + 1.10·13-s − 0.801i·14-s + (0.258 − 0.516i)15-s + 0.250·16-s − 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.270573610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.270573610\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (2 + i)T \) |
| 37 | \( 1 + (6 - i)T \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 5iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.734147090374738336236109399737, −8.610783437927459451696537297198, −7.959606861417185467106879037706, −7.06911250900582908531332821944, −6.12964104100658846555451893656, −5.00294905636482153606728987648, −4.13407548213147831280632398367, −3.84664529118434080072232867263, −2.38792154522695698696898176337, −0.41617444926428710550282091252,
1.89667312577178878410517798237, 2.89766463121113304753673570649, 3.83205948541746557387053060939, 4.97212452293138690184969758999, 5.95474126877153250044088212120, 6.58280040404106115335816011621, 7.55055457524199038899952707646, 8.366668653066126170496621987381, 8.928070521712944586832036579470, 10.50481981018230467959420619220