L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − 3.37i·7-s + i·8-s − 9-s + 0.627·11-s − i·12-s − 2i·13-s − 3.37·14-s + 16-s + 4.74i·17-s + i·18-s + 0.627·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 1.27i·7-s + 0.353i·8-s − 0.333·9-s + 0.189·11-s − 0.288i·12-s − 0.554i·13-s − 0.901·14-s + 0.250·16-s + 1.15i·17-s + 0.235i·18-s + 0.144·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.720502005\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.720502005\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 3.37iT - 7T^{2} \) |
| 11 | \( 1 - 0.627T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 4.74iT - 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 - 3.37iT - 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 37 | \( 1 - 0.744iT - 37T^{2} \) |
| 41 | \( 1 - 0.744T + 41T^{2} \) |
| 43 | \( 1 - 0.627iT - 43T^{2} \) |
| 47 | \( 1 - 6.74iT - 47T^{2} \) |
| 53 | \( 1 - 1.37iT - 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 10.7iT - 67T^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 + 8.11iT - 73T^{2} \) |
| 79 | \( 1 - 4.62T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.192017536045493867950268112862, −7.68984420582537060259168678807, −6.69394176156565045243946232568, −5.91421789864177213277709648311, −4.95636753391204270425545821957, −4.28061554501289653605996187645, −3.63321904503824463639674719434, −2.94818821688529834835537399911, −1.65074247318675913164889480236, −0.66313467354204120915462330711,
0.836845891708379972901534893699, 2.20687777356438528072049199590, 2.86188752524180018574005077648, 4.06590189310268286276452468011, 5.06264363623654079150035824049, 5.50333201654178268739340122615, 6.51575954451201944463177475635, 6.77123638935117069321926595570, 7.69042602770938014286757236363, 8.487683533917226468506617671636