L(s) = 1 | + (−7.29 + 12.6i)5-s + (−18.4 + 0.998i)7-s + (−38.7 + 22.3i)11-s + 76.8i·13-s + (−58.3 − 101. i)17-s + (−83.5 − 48.2i)19-s + (6.88 + 3.97i)23-s + (−43.9 − 76.1i)25-s + 86.8i·29-s + (−216. + 124. i)31-s + (122. − 240. i)35-s + (−160. + 277. i)37-s + 231.·41-s + 413.·43-s + (235. − 407. i)47-s + ⋯ |
L(s) = 1 | + (−0.652 + 1.13i)5-s + (−0.998 + 0.0539i)7-s + (−1.06 + 0.612i)11-s + 1.63i·13-s + (−0.832 − 1.44i)17-s + (−1.00 − 0.582i)19-s + (0.0624 + 0.0360i)23-s + (−0.351 − 0.608i)25-s + 0.556i·29-s + (−1.25 + 0.723i)31-s + (0.590 − 1.16i)35-s + (−0.711 + 1.23i)37-s + 0.881·41-s + 1.46·43-s + (0.730 − 1.26i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1372744928\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1372744928\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (18.4 - 0.998i)T \) |
good | 5 | \( 1 + (7.29 - 12.6i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (38.7 - 22.3i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 76.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (58.3 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (83.5 + 48.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-6.88 - 3.97i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 86.8iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (216. - 124. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (160. - 277. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 231.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-235. + 407. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (600. - 346. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-143. - 249. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-740. - 427. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (240. + 416. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 930. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (98.4 - 56.8i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (111. - 192. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 692.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-258. + 448. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 807. iT - 9.12e5T^{2} \) |
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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
−9.359559675686122651480094597664, −8.838009122001575548643241240598, −7.30010814889898876613491989642, −7.10462254509566620902391911246, −6.34405886000425028584715756160, −4.95740429300396565468142582819, −4.05171492963616846078145900707, −2.95059224365127950750454921462, −2.23103726452747238989183541427, −0.05544491258592391576390275563,
0.62089892148048534327226213811, 2.33985113276575453870867446692, 3.57517387237407283153946580414, 4.29822889424976700404960416446, 5.61135319534140415981377242341, 5.99810865070002466268794758264, 7.43858436949939762860162413236, 8.209375227227839225674666703721, 8.660778429244482699572377026741, 9.701968746618142280342210484825