L(s) = 1 | + 1.69·2-s − 3-s + 0.874·4-s + (2.22 + 0.217i)5-s − 1.69·6-s + (−0.698 − 2.55i)7-s − 1.90·8-s + 9-s + (3.77 + 0.368i)10-s + (3.25 − 0.658i)11-s − 0.874·12-s − 5.21i·13-s + (−1.18 − 4.32i)14-s + (−2.22 − 0.217i)15-s − 4.98·16-s − 3.07i·17-s + ⋯ |
L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.437·4-s + (0.995 + 0.0972i)5-s − 0.692·6-s + (−0.263 − 0.964i)7-s − 0.674·8-s + 0.333·9-s + (1.19 + 0.116i)10-s + (0.980 − 0.198i)11-s − 0.252·12-s − 1.44i·13-s + (−0.316 − 1.15i)14-s + (−0.574 − 0.0561i)15-s − 1.24·16-s − 0.746i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.507704398\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.507704398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (-2.22 - 0.217i)T \) |
| 7 | \( 1 + (0.698 + 2.55i)T \) |
| 11 | \( 1 + (-3.25 + 0.658i)T \) |
good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 13 | \( 1 + 5.21iT - 13T^{2} \) |
| 17 | \( 1 + 3.07iT - 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 6.74iT - 23T^{2} \) |
| 29 | \( 1 - 0.101iT - 29T^{2} \) |
| 31 | \( 1 + 6.52iT - 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.80T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 + 2.30iT - 53T^{2} \) |
| 59 | \( 1 - 7.18iT - 59T^{2} \) |
| 61 | \( 1 + 4.40T + 61T^{2} \) |
| 67 | \( 1 + 4.25iT - 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 + 8.09iT - 73T^{2} \) |
| 79 | \( 1 - 14.2iT - 79T^{2} \) |
| 83 | \( 1 - 7.09iT - 83T^{2} \) |
| 89 | \( 1 - 16.3iT - 89T^{2} \) |
| 97 | \( 1 + 2.98T + 97T^{2} \) |
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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
−9.611531787121296719812596043259, −9.175392521546268516902908427530, −7.65766664921478321476116805798, −6.81761868483110226584204668102, −5.83538355675468566207715166838, −5.63026631061999630723396793097, −4.39115192140928173742191951586, −3.69265181330592123812652427430, −2.52702766398639978837091658439, −0.810383788209496322227718094973,
1.74193175519245445604651517131, 2.75977530059333940030278175556, 4.21368323414137603047421758420, 4.68955499298587525846255490510, 5.82334816771995937447502225602, 6.33136607598058268625118870353, 6.75730383080213394095097026190, 8.758229215409626149681245447842, 8.990998603982932849234185976568, 9.956988604044787246407669433037