L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 4·13-s + 16-s + 4·19-s − 20-s − 8·23-s + 25-s − 4·26-s − 2·29-s − 4·31-s + 32-s − 2·37-s + 4·38-s − 40-s − 10·41-s − 43-s − 8·46-s + 12·47-s − 7·49-s + 50-s − 4·52-s − 2·53-s − 2·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.10·13-s + 1/4·16-s + 0.917·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s − 0.784·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.56·41-s − 0.152·43-s − 1.17·46-s + 1.75·47-s − 49-s + 0.141·50-s − 0.554·52-s − 0.274·53-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.88121062094934296124121687182, −7.36617316577632833814789172233, −6.66752208121222356980996238882, −5.69081600221898085162106772500, −5.12557733247462741530059752750, −4.24718414486006713061233724267, −3.55105542891525590393894273039, −2.63672832248773395308304521780, −1.65432668113079297419751568145, 0,
1.65432668113079297419751568145, 2.63672832248773395308304521780, 3.55105542891525590393894273039, 4.24718414486006713061233724267, 5.12557733247462741530059752750, 5.69081600221898085162106772500, 6.66752208121222356980996238882, 7.36617316577632833814789172233, 7.88121062094934296124121687182