L(s) = 1 | + (2.40 − 0.874i)3-s + (−0.173 + 0.984i)5-s + (−0.118 − 0.204i)7-s + (2.70 − 2.27i)9-s + (2.60 − 4.51i)11-s + (−1.93 − 0.705i)13-s + (0.443 + 2.51i)15-s + (4.62 + 3.88i)17-s + (1.86 + 3.93i)19-s + (−0.462 − 0.387i)21-s + (−0.685 − 3.88i)23-s + (−0.939 − 0.342i)25-s + (0.682 − 1.18i)27-s + (−6.90 + 5.79i)29-s + (−1.86 − 3.23i)31-s + ⋯ |
L(s) = 1 | + (1.38 − 0.504i)3-s + (−0.0776 + 0.440i)5-s + (−0.0446 − 0.0772i)7-s + (0.902 − 0.757i)9-s + (0.786 − 1.36i)11-s + (−0.537 − 0.195i)13-s + (0.114 + 0.649i)15-s + (1.12 + 0.941i)17-s + (0.427 + 0.903i)19-s + (−0.100 − 0.0846i)21-s + (−0.142 − 0.810i)23-s + (−0.187 − 0.0684i)25-s + (0.131 − 0.227i)27-s + (−1.28 + 1.07i)29-s + (−0.335 − 0.580i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.03678 - 0.429106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.03678 - 0.429106i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (-1.86 - 3.93i)T \) |
good | 3 | \( 1 + (-2.40 + 0.874i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (0.118 + 0.204i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.60 + 4.51i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.93 + 0.705i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.62 - 3.88i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.685 + 3.88i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.90 - 5.79i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.86 + 3.23i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + (11.1 - 4.04i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0112 - 0.0636i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.65 - 5.58i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.300 - 1.70i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.134 - 0.112i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.78 + 10.0i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.759 - 0.637i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.860 - 4.87i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (2.27 - 0.829i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.14 + 2.23i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.39 - 4.14i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.51 + 2.00i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (2.70 + 2.27i)T + (16.8 + 95.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.31339959835176038863763420014, −10.23420533653025992283196682530, −9.315278610398933476805718164859, −8.333868222961957038262467793780, −7.82397725323890559996828969308, −6.72714208649912467859708807898, −5.64554975885369966276154009065, −3.70582510743853166119183029380, −3.15846528580881537394034895266, −1.62823592693087152323691328506,
1.91992793169750117884057449540, 3.23915074122860386116354838456, 4.28895901768994878773425827976, 5.28604206070965539478834698093, 7.09269930170388708600225877419, 7.70236761089811863287981064126, 8.892974315276730678010750391603, 9.554952156266562332739116448528, 9.946773398334591196070954045382, 11.60611501026152485328740637723