| L(s) = 1 | + (−0.0944 + 0.898i)2-s + (1.34 + 1.08i)3-s + (1.15 + 0.246i)4-s + (−0.280 + 2.21i)5-s + (−1.10 + 1.10i)6-s + (−1.53 − 2.65i)7-s + (−0.888 + 2.73i)8-s + (0.642 + 2.93i)9-s + (−1.96 − 0.461i)10-s + (0.137 − 1.30i)11-s + (1.29 + 1.58i)12-s + (−0.522 − 4.96i)13-s + (2.52 − 1.12i)14-s + (−2.78 + 2.68i)15-s + (−0.210 − 0.0937i)16-s + (1.36 − 4.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.0667 + 0.635i)2-s + (0.779 + 0.626i)3-s + (0.579 + 0.123i)4-s + (−0.125 + 0.992i)5-s + (−0.450 + 0.453i)6-s + (−0.579 − 1.00i)7-s + (−0.314 + 0.967i)8-s + (0.214 + 0.976i)9-s + (−0.621 − 0.145i)10-s + (0.0414 − 0.393i)11-s + (0.374 + 0.458i)12-s + (−0.144 − 1.37i)13-s + (0.676 − 0.301i)14-s + (−0.719 + 0.694i)15-s + (−0.0526 − 0.0234i)16-s + (0.329 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06214 + 1.19547i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06214 + 1.19547i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.34 - 1.08i)T \) |
| 5 | \( 1 + (0.280 - 2.21i)T \) |
| good | 2 | \( 1 + (0.0944 - 0.898i)T + (-1.95 - 0.415i)T^{2} \) |
| 7 | \( 1 + (1.53 + 2.65i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.137 + 1.30i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.522 + 4.96i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-1.36 + 4.18i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.05 - 3.24i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.20 + 2.31i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (4.40 - 4.88i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-3.64 - 4.05i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-5.90 + 4.29i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (0.166 + 1.58i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (2.24 + 3.88i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.44 - 4.93i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (2.18 + 6.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.368 - 3.50i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.28 + 12.1i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-3.68 - 4.09i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (2.00 + 6.15i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.08 - 3.69i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.88 + 2.09i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (11.8 - 2.50i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-5.99 - 4.35i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (11.3 - 12.6i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−12.66381325548521330693293673413, −11.10657147180566237801444355846, −10.59227872415021754335403951757, −9.697209893614382606021562155487, −8.258629637932877882565768243139, −7.47797959002637696744087251710, −6.72354806022246915148250782847, −5.33804045227273738062557940326, −3.52731358806281804816935788886, −2.82014470491195719156102209214,
1.57523079701583349970282129729, 2.69306614313065217374841988175, 4.16226958104673787479610004330, 5.98206689620837018202799395629, 6.92442880683881999316805127907, 8.187958493335716579689934752554, 9.297274750326861292645510304271, 9.635318819897270900029230742552, 11.43393199756280564594852270814, 12.02285223952170872431948636881
