L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s − 6·17-s + 18-s + 20-s − 4·23-s − 24-s + 25-s − 27-s − 10·29-s − 30-s + 32-s − 6·34-s + 36-s + 6·37-s + 40-s − 2·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.223·20-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.182·30-s + 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.65876092327017311581367771055, −6.95944784660677058204189193250, −6.22470530780161587442986217128, −5.78361856545318371162626294812, −4.90061029674318665140094486065, −4.30720750657693066153280190423, −3.44345763472942283727469852858, −2.33160601209362951770451873362, −1.60614868356799723779016649868, 0,
1.60614868356799723779016649868, 2.33160601209362951770451873362, 3.44345763472942283727469852858, 4.30720750657693066153280190423, 4.90061029674318665140094486065, 5.78361856545318371162626294812, 6.22470530780161587442986217128, 6.95944784660677058204189193250, 7.65876092327017311581367771055