L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.52 + 0.830i)3-s + 1.00i·4-s + (−2.23 − 0.129i)5-s + (−0.487 − 1.66i)6-s + (−2.47 + 2.47i)7-s + (0.707 − 0.707i)8-s + (1.62 + 2.52i)9-s + (1.48 + 1.66i)10-s + 0.319i·11-s + (−0.830 + 1.52i)12-s + (−3.95 − 3.95i)13-s + 3.49·14-s + (−3.28 − 2.04i)15-s − 1.00·16-s + (−4.30 − 4.30i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.877 + 0.479i)3-s + 0.500i·4-s + (−0.998 − 0.0577i)5-s + (−0.199 − 0.678i)6-s + (−0.935 + 0.935i)7-s + (0.250 − 0.250i)8-s + (0.540 + 0.841i)9-s + (0.470 + 0.528i)10-s + 0.0963i·11-s + (−0.239 + 0.438i)12-s + (−1.09 − 1.09i)13-s + 0.935·14-s + (−0.848 − 0.529i)15-s − 0.250·16-s + (−1.04 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0370401 + 0.225253i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0370401 + 0.225253i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.52 - 0.830i)T \) |
| 5 | \( 1 + (2.23 + 0.129i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (2.47 - 2.47i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.319iT - 11T^{2} \) |
| 13 | \( 1 + (3.95 + 3.95i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.30 + 4.30i)T + 17iT^{2} \) |
| 23 | \( 1 + (2.89 - 2.89i)T - 23iT^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 + (-1.42 + 1.42i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-6.45 - 6.45i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.12 + 4.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.15 + 1.15i)T - 53iT^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 + (11.1 - 11.1i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (9.22 + 9.22i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.00iT - 79T^{2} \) |
| 83 | \( 1 + (-2.31 + 2.31i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 + (2.21 - 2.21i)T - 97iT^{2} \) |
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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
−11.12048665574468505107951076675, −9.963183374604174143128901914116, −9.416184312020898818105065850535, −8.698215479190079770988074262586, −7.74488796283724494256954862535, −7.10331204843888866796800522089, −5.39749024914230510037343817795, −4.23428597025294835104932619967, −3.12989840549974088232513594389, −2.45518227955861953644485432152,
0.12733566511542629729485081357, 2.08876438103708554598132715276, 3.68195792917966577254115990226, 4.33400277040361922650208985289, 6.25108866726231655003577178922, 7.11807005110899735417754475476, 7.46504636575831595366103180004, 8.548760812419329372254457398947, 9.250639741981146374125650928843, 10.13513606456822277916224753703