L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·11-s + 13-s + 14-s + 16-s + 17-s + 4·19-s + 2·22-s − 7·23-s − 26-s − 28-s − 29-s + 3·31-s − 32-s − 34-s − 6·37-s − 4·38-s + 3·41-s − 43-s − 2·44-s + 7·46-s + 12·47-s + 49-s + 52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s + 0.426·22-s − 1.45·23-s − 0.196·26-s − 0.188·28-s − 0.185·29-s + 0.538·31-s − 0.176·32-s − 0.171·34-s − 0.986·37-s − 0.648·38-s + 0.468·41-s − 0.152·43-s − 0.301·44-s + 1.03·46-s + 1.75·47-s + 1/7·49-s + 0.138·52-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.296532418800698668032493796347, −7.64817800946497920285515061772, −7.00292082281195566118758927168, −6.02886062140728316323118853364, −5.50662563248420743518416375774, −4.33428105829599150659696726487, −3.35139702087723593517618361152, −2.49900580864402951337675980753, −1.35741366069724757322673710974, 0,
1.35741366069724757322673710974, 2.49900580864402951337675980753, 3.35139702087723593517618361152, 4.33428105829599150659696726487, 5.50662563248420743518416375774, 6.02886062140728316323118853364, 7.00292082281195566118758927168, 7.64817800946497920285515061772, 8.296532418800698668032493796347