L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (2.12 + 0.707i)5-s + (1 − i)7-s + 2.82i·8-s + (1.00 − 3i)10-s + 4.24·11-s + (−2 − 2i)13-s + (−1.41 − 1.41i)14-s + 4.00·16-s + (2.82 + 2.82i)17-s − 6i·19-s + (−4.24 − 1.41i)20-s − 6i·22-s + (−1.41 − 1.41i)23-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + (0.948 + 0.316i)5-s + (0.377 − 0.377i)7-s + 1.00i·8-s + (0.316 − 0.948i)10-s + 1.27·11-s + (−0.554 − 0.554i)13-s + (−0.377 − 0.377i)14-s + 1.00·16-s + (0.685 + 0.685i)17-s − 1.37i·19-s + (−0.948 − 0.316i)20-s − 1.27i·22-s + (−0.294 − 0.294i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18335 - 0.936531i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18335 - 0.936531i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 - 0.707i)T \) |
good | 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + (2 + 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (1.41 + 1.41i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + (8 - 8i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + (2 - 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.41 + 1.41i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + (-7 - 7i)T + 97iT^{2} \) |
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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
−11.16934397940181467557858404560, −10.37385662099004281689860376579, −9.590976167891695297046578668598, −8.839672671274024664361445301912, −7.57622252429694890301132363395, −6.27998122146795823254546000414, −5.15898382064575334971704120403, −3.99511459055492335677291937081, −2.67789938716730651264373725699, −1.33975368803360284147779574736,
1.64285982951218278896048291358, 3.77112795772153319554191979550, 5.05918100560641238842244979596, 5.82424472434253072344866945587, 6.78458012392423622974426439809, 7.81032563822009154568448863843, 9.024387551775653150079229025650, 9.388919750454900806078666453361, 10.40739029864283790823848461543, 11.97734021669073300832724382135