L(s) = 1 | + (0.618 − 1.61i)3-s + (−2.61 + 0.381i)7-s + (−2.23 − 2.00i)9-s + 0.763i·11-s + 1.23·13-s + 4.47i·17-s + 7.23i·19-s + (−1.00 + 4.47i)21-s + 7.70·23-s + (−4.61 + 2.38i)27-s + 4i·29-s + 3.23i·31-s + (1.23 + 0.472i)33-s − 4.47i·37-s + (0.763 − 2.00i)39-s + ⋯ |
L(s) = 1 | + (0.356 − 0.934i)3-s + (−0.989 + 0.144i)7-s + (−0.745 − 0.666i)9-s + 0.230i·11-s + 0.342·13-s + 1.08i·17-s + 1.66i·19-s + (−0.218 + 0.975i)21-s + 1.60·23-s + (−0.888 + 0.458i)27-s + 0.742i·29-s + 0.581i·31-s + (0.215 + 0.0821i)33-s − 0.735i·37-s + (0.122 − 0.320i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.546495986\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.546495986\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.61 - 0.381i)T \) |
good | 11 | \( 1 - 0.763iT - 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 7.23iT - 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 3.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 + 3.23iT - 47T^{2} \) |
| 53 | \( 1 - 9.23T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 9.70iT - 67T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.23iT - 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.753344284551023072506165787204, −8.614947302027073593333169694084, −7.39669281676374325476923712881, −6.92847577412902189138989999356, −6.03625275029429374429318806538, −5.48216836981047142564659497387, −3.90117317781823289163635379550, −3.29128439581028876099922450522, −2.19651429026356791000189151369, −1.10346644482567590620357782126,
0.60227551516113218389795255513, 2.68989665567287636306012150323, 3.06103057029838804284508463737, 4.20328807990138401602338813646, 4.91668335207169902944680757942, 5.82560272898299811605119514571, 6.77558546470840735589612657897, 7.47969394334314951264002897835, 8.606126959202238738622849030423, 9.178097862976415756127339614996