L(s) = 1 | + (0.517 + 1.31i)2-s + (−0.296 + 0.296i)3-s + (−1.46 + 1.36i)4-s + (−0.707 − 0.707i)5-s + (−0.543 − 0.237i)6-s + i·7-s + (−2.55 − 1.22i)8-s + 2.82i·9-s + (0.564 − 1.29i)10-s + (1.34 + 1.34i)11-s + (0.0305 − 0.838i)12-s + (−0.680 + 0.680i)13-s + (−1.31 + 0.517i)14-s + 0.419·15-s + (0.291 − 3.98i)16-s − 6.14·17-s + ⋯ |
L(s) = 1 | + (0.365 + 0.930i)2-s + (−0.171 + 0.171i)3-s + (−0.732 + 0.680i)4-s + (−0.316 − 0.316i)5-s + (−0.222 − 0.0967i)6-s + 0.377i·7-s + (−0.901 − 0.432i)8-s + 0.941i·9-s + (0.178 − 0.409i)10-s + (0.405 + 0.405i)11-s + (0.00882 − 0.242i)12-s + (−0.188 + 0.188i)13-s + (−0.351 + 0.138i)14-s + 0.108·15-s + (0.0728 − 0.997i)16-s − 1.49·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.133856 - 0.829978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.133856 - 0.829978i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.517 - 1.31i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (0.296 - 0.296i)T - 3iT^{2} \) |
| 11 | \( 1 + (-1.34 - 1.34i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.680 - 0.680i)T - 13iT^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 + (3.27 - 3.27i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.61iT - 23T^{2} \) |
| 29 | \( 1 + (2.41 - 2.41i)T - 29iT^{2} \) |
| 31 | \( 1 + 9.71T + 31T^{2} \) |
| 37 | \( 1 + (-8.06 - 8.06i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.60iT - 41T^{2} \) |
| 43 | \( 1 + (1.00 + 1.00i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + (-6.93 - 6.93i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.09 - 2.09i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.63 - 2.63i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.31 - 4.31i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.00iT - 71T^{2} \) |
| 73 | \( 1 - 0.827iT - 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + (2.03 - 2.03i)T - 83iT^{2} \) |
| 89 | \( 1 - 4.46iT - 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−11.31065535655599528029829284975, −10.36386354259903137898597286407, −9.114963140608892238460080606573, −8.570145873290854377589246458554, −7.61072392263089663500326600251, −6.69631868609188229262972066040, −5.75064418714188684634874812350, −4.66086607492189085352027556979, −4.12837638163654126218087841278, −2.35556065839388454573186216014,
0.42211969877561001252335028130, 2.13493806739217240747095718368, 3.55708846405305069859554702803, 4.19153360623029602142326762831, 5.56980260790964801219032278535, 6.50652007593158353276281918004, 7.49097209379081427726214351713, 9.004823567296882274057765455109, 9.310197009703986842524213550810, 10.63288717805438092973427687229