L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−2.14 − 0.616i)5-s + 1.00i·6-s + (0.0344 − 0.0344i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (1.95 − 1.08i)10-s − 0.241i·11-s + (−0.707 − 0.707i)12-s + (−3.88 + 3.88i)13-s + 0.0487i·14-s + (−1.95 + 1.08i)15-s − 1.00·16-s + (−1.35 − 1.35i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.961 − 0.275i)5-s + 0.408i·6-s + (0.0130 − 0.0130i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.618 − 0.342i)10-s − 0.0729i·11-s + (−0.204 − 0.204i)12-s + (−1.07 + 1.07i)13-s + 0.0130i·14-s + (−0.504 + 0.279i)15-s − 0.250·16-s + (−0.329 − 0.329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.484754 + 0.568906i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.484754 + 0.568906i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.14 + 0.616i)T \) |
| 31 | \( 1 + (3.04 - 4.66i)T \) |
good | 7 | \( 1 + (-0.0344 + 0.0344i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.241iT - 11T^{2} \) |
| 13 | \( 1 + (3.88 - 3.88i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.35 + 1.35i)T + 17iT^{2} \) |
| 19 | \( 1 - 8.10iT - 19T^{2} \) |
| 23 | \( 1 + (-2.64 + 2.64i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 37 | \( 1 + (-4.54 - 4.54i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + (-0.329 + 0.329i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.81 - 2.81i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.597 - 0.597i)T - 53iT^{2} \) |
| 59 | \( 1 - 13.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.6iT - 61T^{2} \) |
| 67 | \( 1 + (0.653 - 0.653i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 + (8.64 - 8.64i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + (-0.681 + 0.681i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 + (5.06 - 5.06i)T - 97iT^{2} \) |
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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
−10.10215994048070797506160562535, −9.246575175649278544383591394714, −8.479045531844660291082926897115, −7.83333866133883693021735015400, −7.07426112502054269387208784963, −6.34927337211942231545434978633, −4.97616488666044251716153333622, −4.15894787895137873883472068545, −2.80482828892197529144951946955, −1.33525732679281149071827176907,
0.42220656534641499699842763370, 2.49174185175988006520074336852, 3.19027090299804439474128764041, 4.33162728679160326509850601061, 5.13360669986687411892821954719, 6.77885679723579819455906426449, 7.52016668983136187997586124903, 8.240066132531027356116703248579, 9.048639144514382690330056404664, 9.854298439486563563944210934686