| L(s) = 1 | + (−3.34 + 5.79i)5-s + (12.7 + 13.4i)7-s + (−28.2 + 16.3i)11-s + 67.9i·13-s + (−15.3 − 26.5i)17-s + (21.8 + 12.6i)19-s + (68.6 + 39.6i)23-s + (40.0 + 69.4i)25-s − 109. i·29-s + (−238. + 137. i)31-s + (−120. + 28.8i)35-s + (160. − 277. i)37-s − 184.·41-s − 364.·43-s + (−25.7 + 44.6i)47-s + ⋯ |
| L(s) = 1 | + (−0.299 + 0.518i)5-s + (0.687 + 0.725i)7-s + (−0.775 + 0.447i)11-s + 1.44i·13-s + (−0.218 − 0.378i)17-s + (0.264 + 0.152i)19-s + (0.622 + 0.359i)23-s + (0.320 + 0.555i)25-s − 0.702i·29-s + (−1.38 + 0.797i)31-s + (−0.582 + 0.139i)35-s + (0.711 − 1.23i)37-s − 0.704·41-s − 1.29·43-s + (−0.0799 + 0.138i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0163i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9664713002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9664713002\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-12.7 - 13.4i)T \) |
| good | 5 | \( 1 + (3.34 - 5.79i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (28.2 - 16.3i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 67.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (15.3 + 26.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-21.8 - 12.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-68.6 - 39.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 109. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (238. - 137. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-160. + 277. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 364.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (25.7 - 44.6i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-532. + 307. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (207. + 359. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-411. - 237. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (142. + 246. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 965. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-225. + 130. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (219. - 379. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 76.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-356. + 617. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 410. iT - 9.12e5T^{2} \) |
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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
−9.920927468680886005524608393067, −9.143497891107983129200264692560, −8.388283366387968497184104531881, −7.34877491886656358970039665473, −6.86555384055267406611764388531, −5.56210078239009562707910818000, −4.87563536127266655214740840902, −3.77267995539483372271319578326, −2.56320547791130402071101931522, −1.66418899040243214065446270474,
0.24863499336939913872688196936, 1.22366884395191290331047252109, 2.74957159695832936645223513462, 3.79975717898147737601832152694, 4.90618064841216331309969842548, 5.44623566681436091423143238783, 6.70590772007912753560606402571, 7.76606464515728314031111129953, 8.165139664683448128569403525195, 8.997892852723065353763346916592
