L(s) = 1 | − 2.30·3-s − 2.25·5-s − 2.39·7-s + 2.30·9-s + 11-s − 2.79·13-s + 5.19·15-s + 1.45·17-s − 6.96·19-s + 5.51·21-s + 2.28·23-s + 0.0906·25-s + 1.60·27-s + 4.69·29-s − 2.86·31-s − 2.30·33-s + 5.39·35-s + 5.35·37-s + 6.44·39-s + 1.80·41-s + 5.80·43-s − 5.19·45-s + 5.45·47-s − 1.27·49-s − 3.36·51-s + 3.77·53-s − 2.25·55-s + ⋯ |
L(s) = 1 | − 1.32·3-s − 1.00·5-s − 0.904·7-s + 0.767·9-s + 0.301·11-s − 0.776·13-s + 1.34·15-s + 0.354·17-s − 1.59·19-s + 1.20·21-s + 0.476·23-s + 0.0181·25-s + 0.309·27-s + 0.872·29-s − 0.513·31-s − 0.400·33-s + 0.912·35-s + 0.881·37-s + 1.03·39-s + 0.282·41-s + 0.885·43-s − 0.774·45-s + 0.796·47-s − 0.181·49-s − 0.470·51-s + 0.518·53-s − 0.304·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 + 2.39T + 7T^{2} \) |
| 13 | \( 1 + 2.79T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 + 6.96T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 - 5.35T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 0.632T + 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 - 4.41T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 0.855T + 97T^{2} \) |
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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
−7.43826406222163176842107422366, −7.01472117764977679856740518439, −6.17707446069627460582448402221, −5.80126740018601944609952056761, −4.69869764698034429971578802315, −4.28069248226960069615380681885, −3.35814576622199631288939846630, −2.37408046162847524022008354528, −0.817959950311494957819764603350, 0,
0.817959950311494957819764603350, 2.37408046162847524022008354528, 3.35814576622199631288939846630, 4.28069248226960069615380681885, 4.69869764698034429971578802315, 5.80126740018601944609952056761, 6.17707446069627460582448402221, 7.01472117764977679856740518439, 7.43826406222163176842107422366