L(s) = 1 | + (0.707 − 0.707i)2-s − i·3-s − 1.00i·4-s + (−2.27 − 2.27i)5-s + (−0.707 − 0.707i)6-s + (−2.27 + 1.35i)7-s + (−0.707 − 0.707i)8-s − 9-s − 3.21·10-s + (0.355 + 0.355i)11-s − 1.00·12-s + (−2 + 3i)13-s + (−0.648 + 2.56i)14-s + (−2.27 + 2.27i)15-s − 1.00·16-s + 4.32·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (−1.01 − 1.01i)5-s + (−0.288 − 0.288i)6-s + (−0.858 + 0.512i)7-s + (−0.250 − 0.250i)8-s − 0.333·9-s − 1.01·10-s + (0.107 + 0.107i)11-s − 0.288·12-s + (−0.554 + 0.832i)13-s + (−0.173 + 0.685i)14-s + (−0.586 + 0.586i)15-s − 0.250·16-s + 1.04·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.178308 + 0.642338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.178308 + 0.642338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (2.27 - 1.35i)T \) |
| 13 | \( 1 + (2 - 3i)T \) |
good | 5 | \( 1 + (2.27 + 2.27i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.355 - 0.355i)T + 11iT^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + (5.98 + 5.98i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.38iT - 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + (1.08 + 1.08i)T + 31iT^{2} \) |
| 37 | \( 1 + (5.18 + 5.18i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.53 + 3.53i)T + 41iT^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 + (-4.71 + 4.71i)T - 47iT^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + (3.61 - 3.61i)T - 59iT^{2} \) |
| 61 | \( 1 - 4.32iT - 61T^{2} \) |
| 67 | \( 1 + (0.531 - 0.531i)T - 67iT^{2} \) |
| 71 | \( 1 + (-6.38 + 6.38i)T - 71iT^{2} \) |
| 73 | \( 1 + (-5.18 + 5.18i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + (6.71 + 6.71i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.75 + 3.75i)T - 89iT^{2} \) |
| 97 | \( 1 + (-12.9 - 12.9i)T + 97iT^{2} \) |
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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
−10.42511969371927479012082006140, −9.188309656977777487441998931651, −8.753732384268770195545175088946, −7.49513533358234226230251911659, −6.62964488132516350036814839764, −5.45436000630638271966725043103, −4.48424210418940467974105041290, −3.44832421261165002470313527916, −2.07731796117217031843678152301, −0.31261432854847018296460341329,
3.04060188597766700051696392052, 3.57863053519314764481235666036, 4.58071381670826791247292107013, 5.94871704967425651551681092484, 6.72048394284943791511316786852, 7.70514705300580946801869500574, 8.317068813277294864814961562888, 9.839611261464844290212046206364, 10.40592162694200815416269773188, 11.22119635801516766509966082828