L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s − 8·7-s − 8·8-s − 10·10-s + 20·11-s − 82·13-s + 16·14-s + 16·16-s + 18·17-s + 19·19-s + 20·20-s − 40·22-s + 88·23-s + 25·25-s + 164·26-s − 32·28-s + 186·29-s − 248·31-s − 32·32-s − 36·34-s − 40·35-s + 262·37-s − 38·38-s − 40·40-s − 246·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.431·7-s − 0.353·8-s − 0.316·10-s + 0.548·11-s − 1.74·13-s + 0.305·14-s + 1/4·16-s + 0.256·17-s + 0.229·19-s + 0.223·20-s − 0.387·22-s + 0.797·23-s + 1/5·25-s + 1.23·26-s − 0.215·28-s + 1.19·29-s − 1.43·31-s − 0.176·32-s − 0.181·34-s − 0.193·35-s + 1.16·37-s − 0.162·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 23 | \( 1 - 88 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 262 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 288 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 - 302 T + p^{3} T^{2} \) |
| 59 | \( 1 + 72 T + p^{3} T^{2} \) |
| 61 | \( 1 + 546 T + p^{3} T^{2} \) |
| 67 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 240 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 + 800 T + p^{3} T^{2} \) |
| 83 | \( 1 - 116 T + p^{3} T^{2} \) |
| 89 | \( 1 + 766 T + p^{3} T^{2} \) |
| 97 | \( 1 - 790 T + p^{3} T^{2} \) |
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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
−8.818028357285734934805627889287, −7.66798383320450535999491147879, −7.13286970642483502107067432767, −6.32586406496860525047127791422, −5.41027502682994136960182094748, −4.48609392900545554824330357104, −3.15589957216268453979641664269, −2.36900034725003712004614235216, −1.20195999102220714268644792579, 0,
1.20195999102220714268644792579, 2.36900034725003712004614235216, 3.15589957216268453979641664269, 4.48609392900545554824330357104, 5.41027502682994136960182094748, 6.32586406496860525047127791422, 7.13286970642483502107067432767, 7.66798383320450535999491147879, 8.818028357285734934805627889287