| L(s) = 1 | + (0.174 + 0.793i)2-s + (2.61 + 1.46i)3-s + (3.03 − 1.40i)4-s + (0.375 + 0.104i)5-s + (−0.706 + 2.33i)6-s + (−9.31 + 8.82i)7-s + (3.60 + 4.74i)8-s + (4.69 + 7.67i)9-s + (−0.0171 + 0.316i)10-s + (3.86 − 3.28i)11-s + (9.99 + 0.774i)12-s + (−2.28 + 0.769i)13-s + (−8.62 − 5.85i)14-s + (0.830 + 0.823i)15-s + (5.51 − 6.49i)16-s + (19.9 − 21.0i)17-s + ⋯ |
| L(s) = 1 | + (0.0872 + 0.396i)2-s + (0.872 + 0.488i)3-s + (0.757 − 0.350i)4-s + (0.0751 + 0.0208i)5-s + (−0.117 + 0.388i)6-s + (−1.33 + 1.26i)7-s + (0.450 + 0.593i)8-s + (0.522 + 0.852i)9-s + (−0.00171 + 0.0316i)10-s + (0.351 − 0.298i)11-s + (0.832 + 0.0645i)12-s + (−0.175 + 0.0591i)13-s + (−0.616 − 0.417i)14-s + (0.0553 + 0.0549i)15-s + (0.344 − 0.405i)16-s + (1.17 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.86035 + 1.27063i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86035 + 1.27063i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.61 - 1.46i)T \) |
| 59 | \( 1 + (-58.5 - 7.49i)T \) |
| good | 2 | \( 1 + (-0.174 - 0.793i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (-0.375 - 0.104i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (9.31 - 8.82i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (-3.86 + 3.28i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (2.28 - 0.769i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (-19.9 + 21.0i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (-8.57 - 21.5i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (3.94 + 36.3i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (2.06 - 9.39i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (1.21 - 3.03i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (32.2 + 24.4i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (-1.35 + 12.4i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (-35.3 + 41.6i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (68.9 - 19.1i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (17.0 - 0.922i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (32.7 - 7.21i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (72.6 - 55.2i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (-84.7 + 23.5i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (-29.6 + 43.7i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (-2.42 - 14.7i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (74.7 - 39.6i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (-26.5 + 120. i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (-54.9 - 81.0i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−12.55634980717976664806743064623, −11.84341110575563182936175352655, −10.30956477310919868658398292321, −9.669881618871924391873812550185, −8.655693292585086882298022723895, −7.46189734086281639914800181588, −6.26014264491978662253372679233, −5.32027018490026391715400391261, −3.37229029303004821397393847540, −2.33942956050230217277666304254,
1.43371407623170008964830124461, 3.18859092325980411720137171150, 3.81010559203406274320005895581, 6.28771754052470142407073252088, 7.19421485760622737275510877545, 7.85196832932187000646935467314, 9.578036812519560874372701726578, 10.05829683588477287571989162910, 11.40651512385210709320566363445, 12.50242286227019928418252217404
