L(s) = 1 | + (0.707 − 0.707i)2-s + (1.59 − 0.670i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (0.655 − 1.60i)6-s + 3.28·7-s + (−0.707 − 0.707i)8-s + (2.10 − 2.14i)9-s + 1.00·10-s + 1.74·11-s + (−0.670 − 1.59i)12-s + (−3.25 + 3.25i)13-s + (2.32 − 2.32i)14-s + (1.60 + 0.655i)15-s − 1.00·16-s + (1.24 + 1.24i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.922 − 0.387i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.267 − 0.654i)6-s + 1.24·7-s + (−0.250 − 0.250i)8-s + (0.700 − 0.713i)9-s + 0.316·10-s + 0.526·11-s + (−0.193 − 0.461i)12-s + (−0.903 + 0.903i)13-s + (0.621 − 0.621i)14-s + (0.414 + 0.169i)15-s − 0.250·16-s + (0.301 + 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.451695950\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.451695950\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.59 + 0.670i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (4.68 + 3.87i)T \) |
good | 7 | \( 1 - 3.28T + 7T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 + (3.25 - 3.25i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.24 - 1.24i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.258 + 0.258i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.64 + 1.64i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.378 - 0.378i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.64 - 3.64i)T + 31iT^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + (4.35 - 4.35i)T - 43iT^{2} \) |
| 47 | \( 1 + 2.28iT - 47T^{2} \) |
| 53 | \( 1 + 9.35iT - 53T^{2} \) |
| 59 | \( 1 + (-0.666 - 0.666i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.134 + 0.134i)T + 61iT^{2} \) |
| 67 | \( 1 - 8.70iT - 67T^{2} \) |
| 71 | \( 1 + 7.14iT - 71T^{2} \) |
| 73 | \( 1 - 13.7iT - 73T^{2} \) |
| 79 | \( 1 + (-6.38 + 6.38i)T - 79iT^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 + (0.495 - 0.495i)T - 89iT^{2} \) |
| 97 | \( 1 + (-4.16 + 4.16i)T - 97iT^{2} \) |
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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
−9.799728933669368203938124836323, −8.867841922890621450862999294681, −8.177660851585142596115304168294, −7.14833948582230292113597927357, −6.50420862949168980933606384091, −5.17255407653881471024439310601, −4.36326918118867193212402848103, −3.35787456095756801593287083343, −2.17163680407000557121323332794, −1.53162041888292917389456426927,
1.65828813037872823063131484063, 2.81920876902948171902001695147, 3.93158073522232976295317766337, 4.90558403972464999362190554897, 5.34688827315210794646123338513, 6.71993809617379448976448021174, 7.78579540089630700397021115155, 8.096578731977027237724484370577, 9.020873509979568315478418326932, 9.845027585351226454596321629198