L(s) = 1 | + (−1.79 + 1.79i)2-s + (−1.41 + 1.41i)3-s − 4.42i·4-s + (1.69 − 1.46i)5-s − 5.08i·6-s + (−1.27 + 1.27i)7-s + (4.35 + 4.35i)8-s − 1.01i·9-s + (−0.412 + 5.65i)10-s + (6.27 + 6.27i)12-s + (1.59 + 1.59i)13-s − 4.58i·14-s + (−0.326 + 4.47i)15-s − 6.74·16-s + (−2.37 + 2.37i)17-s + (1.82 + 1.82i)18-s + ⋯ |
L(s) = 1 | + (−1.26 + 1.26i)2-s + (−0.818 + 0.818i)3-s − 2.21i·4-s + (0.756 − 0.653i)5-s − 2.07i·6-s + (−0.483 + 0.483i)7-s + (1.53 + 1.53i)8-s − 0.339i·9-s + (−0.130 + 1.78i)10-s + (1.81 + 1.81i)12-s + (0.442 + 0.442i)13-s − 1.22i·14-s + (−0.0842 + 1.15i)15-s − 1.68·16-s + (−0.576 + 0.576i)17-s + (0.430 + 0.430i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0837 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0837 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.102411 - 0.111381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.102411 - 0.111381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.69 + 1.46i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.79 - 1.79i)T - 2iT^{2} \) |
| 3 | \( 1 + (1.41 - 1.41i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.27 - 1.27i)T - 7iT^{2} \) |
| 13 | \( 1 + (-1.59 - 1.59i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.37 - 2.37i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.30T + 19T^{2} \) |
| 23 | \( 1 + (3.16 - 3.16i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 + (2.26 + 2.26i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.30iT - 41T^{2} \) |
| 43 | \( 1 + (4.55 + 4.55i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.46 - 5.46i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.50 + 4.50i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.3iT - 59T^{2} \) |
| 61 | \( 1 + 9.31iT - 61T^{2} \) |
| 67 | \( 1 + (2.46 + 2.46i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + (6.67 + 6.67i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 + (7.39 + 7.39i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.85iT - 89T^{2} \) |
| 97 | \( 1 + (0.550 + 0.550i)T + 97iT^{2} \) |
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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
−10.73424006952241749406762232788, −10.20772987441660082597551373158, −9.273756781601100908481783903494, −8.966410310529786290982105588253, −7.909827616488855223877452281391, −6.67658931153900159993882792451, −5.86148651874757396854560109566, −5.46735505535172340370692446726, −4.25399630302516164401569534992, −1.77803495972838070223512864559,
0.14262903809986750019035881046, 1.49806145286954503167678967047, 2.62222292493268196416977320734, 3.77928076170243773191810474225, 5.67821744839545232310469684592, 6.72148523756098971057142380052, 7.30085494695327749798987552819, 8.488659751564179440494823013730, 9.423999522216587431662225726300, 10.19990587334513148571602556193