L(s) = 1 | + (−0.754 + 2.32i)2-s + (−0.809 − 0.587i)3-s + (−3.19 − 2.32i)4-s + (1.97 − 1.43i)6-s + 3.44·7-s + (3.85 − 2.80i)8-s + (0.309 + 0.951i)9-s + (−1.00 + 3.10i)11-s + (1.22 + 3.76i)12-s + (0.998 + 3.07i)13-s + (−2.59 + 7.98i)14-s + (1.15 + 3.54i)16-s + (−4.08 + 2.97i)17-s − 2.44·18-s + (2.49 − 1.81i)19-s + ⋯ |
L(s) = 1 | + (−0.533 + 1.64i)2-s + (−0.467 − 0.339i)3-s + (−1.59 − 1.16i)4-s + (0.805 − 0.585i)6-s + 1.30·7-s + (1.36 − 0.991i)8-s + (0.103 + 0.317i)9-s + (−0.304 + 0.936i)11-s + (0.352 + 1.08i)12-s + (0.277 + 0.852i)13-s + (−0.693 + 2.13i)14-s + (0.288 + 0.887i)16-s + (−0.991 + 0.720i)17-s − 0.575·18-s + (0.571 − 0.415i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.122785 + 0.776581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.122785 + 0.776581i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.754 - 2.32i)T + (-1.61 - 1.17i)T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 + (1.00 - 3.10i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.998 - 3.07i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (4.08 - 2.97i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.49 + 1.81i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.478 - 1.47i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.52 - 1.83i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.02 - 4.37i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.77 - 5.47i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.67 - 5.15i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 + (-5.72 - 4.15i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (8.21 + 5.96i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.534 - 1.64i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.42 + 7.45i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (1.49 - 1.08i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.577 - 0.419i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.581 + 1.78i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-10.7 - 7.83i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.20 + 2.32i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.63 + 8.11i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (8.61 + 6.26i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.63904435636105470228477086839, −10.83276727668626010971949640918, −9.620466115940727550566203601983, −8.679863367570320098269180114100, −7.892653497319691289688565136194, −7.09344431427846453837093868902, −6.33373754613487371077156553922, −5.11933241249149869451873345905, −4.54088440879824562093802416833, −1.67087400574516844321469803076,
0.71642549981840436284931786188, 2.27538233188217695799508256650, 3.59986952673953312358041495062, 4.71158126189251017707235978082, 5.77370624478958172680785284077, 7.64684852159460881852551039784, 8.532164045079021955877055458722, 9.322431802452453567976710986333, 10.46032761936432033143445327055, 10.99072259720016200221430153128