| L(s) = 1 | + (1.14 + 0.826i)2-s + (−0.213 − 1.89i)3-s + (0.632 + 1.89i)4-s + (2.71 + 1.30i)5-s + (1.32 − 2.35i)6-s + (−1.72 − 1.37i)7-s + (−0.842 + 2.69i)8-s + (−0.622 + 0.142i)9-s + (2.03 + 3.74i)10-s + (−0.325 − 0.518i)11-s + (3.46 − 1.60i)12-s + (−0.132 + 0.580i)13-s + (−0.840 − 2.99i)14-s + (1.89 − 5.42i)15-s + (−3.19 + 2.40i)16-s + (2.96 − 2.96i)17-s + ⋯ |
| L(s) = 1 | + (0.811 + 0.584i)2-s + (−0.123 − 1.09i)3-s + (0.316 + 0.948i)4-s + (1.21 + 0.584i)5-s + (0.539 − 0.959i)6-s + (−0.650 − 0.519i)7-s + (−0.297 + 0.954i)8-s + (−0.207 + 0.0473i)9-s + (0.642 + 1.18i)10-s + (−0.0981 − 0.156i)11-s + (0.999 − 0.463i)12-s + (−0.0367 + 0.160i)13-s + (−0.224 − 0.801i)14-s + (0.489 − 1.40i)15-s + (−0.799 + 0.600i)16-s + (0.719 − 0.719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98462 + 0.249978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98462 + 0.249978i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.14 - 0.826i)T \) |
| 29 | \( 1 + (5.33 - 0.729i)T \) |
| good | 3 | \( 1 + (0.213 + 1.89i)T + (-2.92 + 0.667i)T^{2} \) |
| 5 | \( 1 + (-2.71 - 1.30i)T + (3.11 + 3.90i)T^{2} \) |
| 7 | \( 1 + (1.72 + 1.37i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (0.325 + 0.518i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (0.132 - 0.580i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.96 + 2.96i)T - 17iT^{2} \) |
| 19 | \( 1 + (5.27 + 0.594i)T + (18.5 + 4.22i)T^{2} \) |
| 23 | \( 1 + (-2.42 - 5.02i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (-0.679 - 1.94i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (5.13 + 3.22i)T + (16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (3.72 + 3.72i)T + 41iT^{2} \) |
| 43 | \( 1 + (3.24 - 9.27i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (1.33 + 2.12i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (1.37 - 2.85i)T + (-33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 + (10.8 - 1.22i)T + (59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (-14.6 + 3.34i)T + (60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (0.289 - 1.27i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-15.6 - 5.47i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-5.98 + 9.53i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (4.66 + 5.84i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-8.54 + 2.99i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (1.24 - 11.0i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−12.68268002567143197240368464453, −11.57038039178414951005244329804, −10.38981567768351820086011940395, −9.260245510673787998213506335498, −7.75143846036463304490656246490, −6.86396773954544640301052499023, −6.35027607678793476929459611221, −5.27862045477329061126564355434, −3.46197746703631937406210761236, −2.06787090096705570838195386463,
2.03197151547971508866598054391, 3.57087548882307108714610673151, 4.81092232492174227394566105094, 5.61611383165557238329636771809, 6.52049848592590584395378153097, 8.724344784648803701980267892360, 9.702880017602851667874363119050, 10.14524478395910786848861855238, 11.01435596548015261743626877630, 12.53167057444159754575371813041
