| L(s) = 1 | − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 5·7-s − 8-s + 9-s − 10-s − 3·11-s − 2·12-s − 3·13-s + 5·14-s − 2·15-s + 16-s − 18-s + 20-s + 10·21-s + 3·22-s + 5·23-s + 2·24-s − 4·25-s + 3·26-s + 4·27-s − 5·28-s + 6·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.577·12-s − 0.832·13-s + 1.33·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 2.18·21-s + 0.639·22-s + 1.04·23-s + 0.408·24-s − 4/5·25-s + 0.588·26-s + 0.769·27-s − 0.944·28-s + 1.11·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.65219688253045375375964939542, −11.93416893925429768236807995738, −10.48908165651655127989947106593, −10.06476543076194019396320979498, −8.921814354406228321446481309533, −7.17244428635084092377211487034, −6.31279402094825929172906511169, −5.28324013117968103958478899880, −2.91750389718109755845418463686, 0,
2.91750389718109755845418463686, 5.28324013117968103958478899880, 6.31279402094825929172906511169, 7.17244428635084092377211487034, 8.921814354406228321446481309533, 10.06476543076194019396320979498, 10.48908165651655127989947106593, 11.93416893925429768236807995738, 12.65219688253045375375964939542
