| L(s) = 1 | + (0.0492 + 0.151i)2-s + (6.45 − 4.68i)4-s + (9.69 − 5.56i)5-s + 29.3·7-s + (2.06 + 1.49i)8-s + (1.32 + 1.19i)10-s + (10.1 + 31.2i)11-s + (−9.02 + 27.7i)13-s + (1.44 + 4.45i)14-s + (19.5 − 60.2i)16-s + (−103. − 75.3i)17-s + (3.24 + 2.36i)19-s + (36.4 − 81.3i)20-s + (−4.24 + 3.08i)22-s + (39.8 + 122. i)23-s + ⋯ |
| L(s) = 1 | + (0.0174 + 0.0536i)2-s + (0.806 − 0.585i)4-s + (0.867 − 0.497i)5-s + 1.58·7-s + (0.0910 + 0.0661i)8-s + (0.0417 + 0.0378i)10-s + (0.278 + 0.857i)11-s + (−0.192 + 0.592i)13-s + (0.0276 + 0.0849i)14-s + (0.306 − 0.941i)16-s + (−1.47 − 1.07i)17-s + (0.0392 + 0.0284i)19-s + (0.407 − 0.909i)20-s + (−0.0411 + 0.0298i)22-s + (0.361 + 1.11i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.78215 - 0.660565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.78215 - 0.660565i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-9.69 + 5.56i)T \) |
| good | 2 | \( 1 + (-0.0492 - 0.151i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 29.3T + 343T^{2} \) |
| 11 | \( 1 + (-10.1 - 31.2i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (9.02 - 27.7i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (103. + 75.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-3.24 - 2.36i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-39.8 - 122. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (178. - 129. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-139. - 101. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-64.2 + 197. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-78.0 + 240. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 81.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + (281. - 204. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (324. - 236. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-196. + 604. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (111. + 341. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (72.2 + 52.4i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-434. + 315. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-28.1 - 86.7i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (1.07e3 - 782. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (148. + 107. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (179. + 552. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.34e3 - 975. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.45610295662194218567263564054, −11.01109095891323770181146860023, −9.701285495512236821889875578158, −8.973481925866508022265391937837, −7.53843752922712900685598005751, −6.67219754102745799549839112705, −5.29633245171321528961936365696, −4.67347978168883132822069303292, −2.20064065017969718506523228915, −1.47397179901387886460239700773,
1.67323071529636162596525106323, 2.73385370780458085811063222562, 4.36066682856717485852652841679, 5.84080211545912281563936931153, 6.71909023478751741574446428108, 7.998227591272663908549416763740, 8.635295349160125165449713175000, 10.22812539376133643064676093115, 11.14902537683611837743126520026, 11.46095018633801593209926228153
