| L(s) = 1 | + (−4.10 + 2.37i)2-s + (−2.59 − 1.5i)3-s + (7.23 − 12.5i)4-s + (−2.66 + 10.8i)5-s + 14.2·6-s + (15.6 + 9.85i)7-s + 30.7i·8-s + (4.5 + 7.79i)9-s + (−14.8 − 50.8i)10-s + (−18.3 + 31.7i)11-s + (−37.6 + 21.7i)12-s + 11.1i·13-s + (−87.7 − 3.28i)14-s + (23.2 − 24.2i)15-s + (−14.9 − 25.8i)16-s + (−31.6 − 18.2i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 + 0.838i)2-s + (−0.499 − 0.288i)3-s + (0.904 − 1.56i)4-s + (−0.238 + 0.971i)5-s + 0.967·6-s + (0.846 + 0.532i)7-s + 1.35i·8-s + (0.166 + 0.288i)9-s + (−0.468 − 1.60i)10-s + (−0.502 + 0.871i)11-s + (−0.904 + 0.522i)12-s + 0.237i·13-s + (−1.67 − 0.0627i)14-s + (0.399 − 0.416i)15-s + (−0.232 − 0.403i)16-s + (−0.451 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.648i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.760 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0658604 - 0.178757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0658604 - 0.178757i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.59 + 1.5i)T \) |
| 5 | \( 1 + (2.66 - 10.8i)T \) |
| 7 | \( 1 + (-15.6 - 9.85i)T \) |
| good | 2 | \( 1 + (4.10 - 2.37i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (18.3 - 31.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 11.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (31.6 + 18.2i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (75.6 + 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-59.1 + 34.1i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (130. - 226. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (166. - 95.8i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (259. - 149. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (362. + 209. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-214. + 371. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (390. + 676. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-794. - 458. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 263.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-276. - 159. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-369. - 639. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 182. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (446. + 772. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 311. iT - 9.12e5T^{2} \) |
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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
−14.42674438255699019008797703503, −12.75080597510435120562988121106, −11.22003222481252968838686493676, −10.78725941977778345284932889969, −9.452001539954740687539891639722, −8.332763262532059456444222424122, −7.22337535187071123177554322469, −6.62277209307107747638187005409, −5.03963104866880400130601541325, −2.07787777184753756472478721128,
0.17994573606956475958737837855, 1.61616369707741422053923794402, 3.92704982632993507920105541262, 5.56261843293285131124500753597, 7.69381067590784318331038183218, 8.388641684733610130329247248743, 9.440021082057126009102905205836, 10.66437698545977545725101895869, 11.16637836468922823273426659964, 12.22076669796850735471423557301
