L(s) = 1 | + (1.31 − 2.50i)2-s + (−0.0469 + 0.0812i)3-s + (−4.56 − 6.57i)4-s + (12.4 − 7.20i)5-s + (0.142 + 0.224i)6-s + (−15.0 + 10.7i)7-s + (−22.4 + 2.80i)8-s + (13.4 + 23.3i)9-s + (−1.69 − 40.7i)10-s + (31.6 + 18.2i)11-s + (0.748 − 0.0621i)12-s + 15.3i·13-s + (7.22 + 51.8i)14-s + 1.35i·15-s + (−22.4 + 59.9i)16-s + (−54.1 − 31.2i)17-s + ⋯ |
L(s) = 1 | + (0.463 − 0.886i)2-s + (−0.00903 + 0.0156i)3-s + (−0.570 − 0.821i)4-s + (1.11 − 0.644i)5-s + (0.00967 + 0.0152i)6-s + (−0.813 + 0.581i)7-s + (−0.992 + 0.124i)8-s + (0.499 + 0.865i)9-s + (−0.0534 − 1.28i)10-s + (0.866 + 0.500i)11-s + (0.0180 − 0.00149i)12-s + 0.326i·13-s + (0.137 + 0.990i)14-s + 0.0232i·15-s + (−0.350 + 0.936i)16-s + (−0.772 − 0.446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.20422 - 0.881980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20422 - 0.881980i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.31 + 2.50i)T \) |
| 7 | \( 1 + (15.0 - 10.7i)T \) |
good | 3 | \( 1 + (0.0469 - 0.0812i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (-12.4 + 7.20i)T + (62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-31.6 - 18.2i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 15.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (54.1 + 31.2i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (34.6 + 60.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (12.3 - 7.14i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 157.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-165. + 286. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (38.0 + 65.9i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 335. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 484. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (187. + 324. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-70.8 + 122. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (159. - 277. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-213. + 123. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (178. + 102. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 82.4iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (575. + 331. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-564. + 325. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 790.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-631. + 364. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 386. iT - 9.12e5T^{2} \) |
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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
−16.56636606066505257485143775608, −15.04958076376598776670888628257, −13.45835115822877280089313678552, −13.00880054062869549733425332062, −11.52939587483718343998571576102, −9.846814639992406271919361845794, −9.157450810184317133882339328307, −6.24864042056805407007202567573, −4.67505306978126347346995497015, −2.09715622940824625987543486802,
3.67584833602192963673360584173, 6.11005944561243325902106307348, 6.85147212231403325745140760337, 9.014652505318608512941780497396, 10.28204850126793859381287218075, 12.41546844120904551867741543076, 13.56100333459441254423671190270, 14.44474183166156132261286305121, 15.68662915373557808286560849593, 17.02775040546662327471014385593