| L(s) = 1 | + (−0.778 − 1.57i)3-s + (−0.245 + 0.215i)5-s + (2.04 + 1.68i)7-s + (−0.0583 + 0.0760i)9-s + (1.80 − 0.118i)11-s + (1.25 + 1.25i)13-s + (0.530 + 0.219i)15-s + (3.87 − 1.42i)17-s + (−0.905 − 6.87i)19-s + (1.06 − 4.53i)21-s + (−1.88 − 0.930i)23-s + (−0.638 + 4.85i)25-s + (−5.01 − 0.996i)27-s + (−1.09 − 5.48i)29-s + (8.63 − 4.26i)31-s + ⋯ |
| L(s) = 1 | + (−0.449 − 0.911i)3-s + (−0.109 + 0.0962i)5-s + (0.771 + 0.636i)7-s + (−0.0194 + 0.0253i)9-s + (0.544 − 0.0357i)11-s + (0.348 + 0.348i)13-s + (0.137 + 0.0567i)15-s + (0.938 − 0.344i)17-s + (−0.207 − 1.57i)19-s + (0.233 − 0.988i)21-s + (−0.393 − 0.193i)23-s + (−0.127 + 0.970i)25-s + (−0.964 − 0.191i)27-s + (−0.202 − 1.01i)29-s + (1.55 − 0.765i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20741 - 0.640186i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20741 - 0.640186i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.04 - 1.68i)T \) |
| 17 | \( 1 + (-3.87 + 1.42i)T \) |
| good | 3 | \( 1 + (0.778 + 1.57i)T + (-1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (0.245 - 0.215i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (-1.80 + 0.118i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 1.25i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.905 + 6.87i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (1.88 + 0.930i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (1.09 + 5.48i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-8.63 + 4.26i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.198 - 3.03i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.328 - 1.65i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.913 + 2.20i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-2.97 + 11.1i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.14 - 6.70i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (12.7 + 1.68i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.98 - 5.84i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-10.0 - 5.81i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.85 - 5.24i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.37 - 0.805i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (6.74 - 13.6i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-2.37 + 5.72i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (11.8 + 3.18i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.250 - 0.0498i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.40703318780641815273370485115, −9.934673521424192055661601084816, −8.988563321975525976919270419964, −8.038423743619118709742499857308, −7.12836232276823458697690702682, −6.29742505719229240681258559079, −5.34546350931864081305460400931, −4.10827477547675232317643051609, −2.45504824771687283892025577051, −1.08530137588269465901944856420,
1.46364789025605857205374194974, 3.58450377052160589329851949389, 4.36423148433834064531612060611, 5.32544861370020067422772133071, 6.32991006121924423506982684940, 7.72629194619060929764306282541, 8.316039907883745536795413917830, 9.650540749160108826720958821978, 10.38614401608537752344693947370, 10.87413218569863806656053098865
