| L(s) = 1 | + (−9.31 − 3.02i)3-s + (−11.0 + 1.81i)5-s + 0.760i·7-s + (55.7 + 40.5i)9-s + (50.6 − 36.7i)11-s + (−24.8 + 34.1i)13-s + (108. + 16.4i)15-s + (−10.7 + 3.48i)17-s + (−0.485 − 1.49i)19-s + (2.30 − 7.08i)21-s + (110. + 152. i)23-s + (118. − 40.0i)25-s + (−241. − 332. i)27-s + (−77.4 + 238. i)29-s + (−31.0 − 95.5i)31-s + ⋯ |
| L(s) = 1 | + (−1.79 − 0.582i)3-s + (−0.986 + 0.162i)5-s + 0.0410i·7-s + (2.06 + 1.50i)9-s + (1.38 − 1.00i)11-s + (−0.530 + 0.729i)13-s + (1.86 + 0.283i)15-s + (−0.153 + 0.0497i)17-s + (−0.00586 − 0.0180i)19-s + (0.0239 − 0.0735i)21-s + (1.00 + 1.38i)23-s + (0.947 − 0.320i)25-s + (−1.72 − 2.36i)27-s + (−0.495 + 1.52i)29-s + (−0.179 − 0.553i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.621741 + 0.142988i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621741 + 0.142988i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (11.0 - 1.81i)T \) |
| good | 3 | \( 1 + (9.31 + 3.02i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 - 0.760iT - 343T^{2} \) |
| 11 | \( 1 + (-50.6 + 36.7i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (24.8 - 34.1i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (10.7 - 3.48i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (0.485 + 1.49i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-110. - 152. i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (77.4 - 238. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (31.0 + 95.5i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-55.9 + 77.0i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-253. - 184. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 188. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-59.9 - 19.4i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-91.5 - 29.7i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-173. - 125. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (144. - 105. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-153. + 49.8i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (270. - 831. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-354. - 487. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (49.5 - 152. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-757. + 246. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (577. - 419. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-773. - 251. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−13.11753429452029394180452093119, −12.04374669262199257850327349030, −11.46829641617644374262540656593, −10.85953013062137403439727420846, −9.133955836398586711886269964549, −7.40236988732377601465877865630, −6.66783396780247663083999954292, −5.43475549355067543842582733316, −4.02322779202894538595986440141, −1.04859412007670465192157969558,
0.61729397874779643776210738966, 4.08870797558470468340730581494, 4.88530727344447822790238992708, 6.36744359800354009156681240997, 7.35982239809625391896246424077, 9.231098991203635676617855845587, 10.36216593200053861161471096501, 11.30095278181358903567973099033, 12.12764209699629766052197330597, 12.67611780814329644269998210395
