L(s) = 1 | + (−1.29 − 0.747i)2-s + (0.118 + 0.204i)4-s + i·5-s + (−4.18 + 2.41i)7-s + 2.63i·8-s + (0.747 − 1.29i)10-s + (0.926 + 0.534i)11-s + (0.331 − 3.59i)13-s + 7.21·14-s + (2.20 − 3.82i)16-s + (−1.77 − 3.08i)17-s + (4.96 − 2.86i)19-s + (−0.204 + 0.118i)20-s + (−0.799 − 1.38i)22-s + (3.54 − 6.13i)23-s + ⋯ |
L(s) = 1 | + (−0.915 − 0.528i)2-s + (0.0591 + 0.102i)4-s + 0.447i·5-s + (−1.57 + 0.912i)7-s + 0.932i·8-s + (0.236 − 0.409i)10-s + (0.279 + 0.161i)11-s + (0.0918 − 0.995i)13-s + 1.92·14-s + (0.552 − 0.956i)16-s + (−0.431 − 0.747i)17-s + (1.13 − 0.657i)19-s + (−0.0458 + 0.0264i)20-s + (−0.170 − 0.295i)22-s + (0.738 − 1.27i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0791 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.377554 - 0.408706i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.377554 - 0.408706i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-0.331 + 3.59i)T \) |
good | 2 | \( 1 + (1.29 + 0.747i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (4.18 - 2.41i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.926 - 0.534i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.77 + 3.08i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.96 + 2.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.54 + 6.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.736 + 1.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (0.0219 + 0.0126i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.232 - 0.133i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.77 - 3.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.51iT - 47T^{2} \) |
| 53 | \( 1 + 0.991T + 53T^{2} \) |
| 59 | \( 1 + (-7.55 + 4.36i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.48 + 2.58i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.72 + 3.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 8.78T + 79T^{2} \) |
| 83 | \( 1 - 0.725iT - 83T^{2} \) |
| 89 | \( 1 + (11.6 + 6.75i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.97 - 1.71i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.32542587436769251292312545029, −9.441633458089639610586145942165, −9.194623269228530187602184290075, −8.058473425464262052945863164299, −6.89958304849212400260996847239, −6.02199659092725803318745653811, −4.99447724306036562433896379932, −3.13019511639156541836219237192, −2.54575940764480281079593014839, −0.50982224875709423994697132729,
1.13585218698366801377739690055, 3.43010500714002175036193578080, 4.07913668123172208420083581093, 5.76211935693528320534137809776, 6.83272029632000892803359438388, 7.22571152334745818022516173754, 8.399307805216956084477784901024, 9.280882050891246185369968102000, 9.699225877242514651862698031844, 10.55548559452901491406816626928