L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 6·11-s + 2·13-s − 14-s + 16-s − 2·17-s + 4·19-s − 6·22-s − 4·23-s + 2·26-s − 28-s − 2·29-s − 2·31-s + 32-s − 2·34-s − 10·37-s + 4·38-s − 6·41-s − 2·43-s − 6·44-s − 4·46-s + 2·47-s + 49-s + 2·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 1.27·22-s − 0.834·23-s + 0.392·26-s − 0.188·28-s − 0.371·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s − 1.64·37-s + 0.648·38-s − 0.937·41-s − 0.304·43-s − 0.904·44-s − 0.589·46-s + 0.291·47-s + 1/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.217587117868143369543496852139, −7.43216137868225182671404448289, −6.82159598408071217496958775519, −5.72732538785287537386713703751, −5.40227581520164199067646921877, −4.43838092100299059694787211741, −3.47230799131228886112245141292, −2.77431134095133333097439110965, −1.74800605557638450377762078020, 0,
1.74800605557638450377762078020, 2.77431134095133333097439110965, 3.47230799131228886112245141292, 4.43838092100299059694787211741, 5.40227581520164199067646921877, 5.72732538785287537386713703751, 6.82159598408071217496958775519, 7.43216137868225182671404448289, 8.217587117868143369543496852139