L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.707 + 0.707i)3-s + (0.866 + 0.499i)4-s + (−1.22 − 1.22i)5-s + (0.866 − 0.500i)6-s − i·7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (0.866 + 1.49i)10-s + (0.707 + 0.707i)11-s + (−0.965 + 0.258i)12-s + (−0.366 + 0.366i)13-s + (−0.258 + 0.965i)14-s + 1.73·15-s + (0.500 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.707 + 0.707i)3-s + (0.866 + 0.499i)4-s + (−1.22 − 1.22i)5-s + (0.866 − 0.500i)6-s − i·7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (0.866 + 1.49i)10-s + (0.707 + 0.707i)11-s + (−0.965 + 0.258i)12-s + (−0.366 + 0.366i)13-s + (−0.258 + 0.965i)14-s + 1.73·15-s + (0.500 + 0.866i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3846003222\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3846003222\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
good | 5 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 13 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - 1.93T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.774163689661369185423440592102, −8.165433317139602992155868064586, −7.28666197069968562097276048366, −6.82417320872108960930365000574, −5.73769980910223247850783044723, −4.66162652594714173620312044567, −3.93826934619315182441673716256, −3.73807585504115601458433168742, −1.78937204770319098185640147088, −0.73920905033876603328891847385,
0.53012419786571848095327523369, 2.23128279064347330851027903244, 2.75850407653702511671254820587, 3.99125758522670388942875904521, 5.27199729035796345586345788002, 6.05520081442777269604329021064, 6.75230925054840958158873410696, 7.06391350212354678381919213998, 7.997153866278728595547961213562, 8.436271010862283010868977961889