L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·11-s − 7·13-s − 14-s + 16-s − 7·17-s + 8·19-s + 2·22-s − 5·23-s − 7·26-s − 28-s − 9·29-s + 31-s + 32-s − 7·34-s + 2·37-s + 8·38-s − 11·41-s − 3·43-s + 2·44-s − 5·46-s + 4·47-s + 49-s − 7·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.603·11-s − 1.94·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s + 1.83·19-s + 0.426·22-s − 1.04·23-s − 1.37·26-s − 0.188·28-s − 1.67·29-s + 0.179·31-s + 0.176·32-s − 1.20·34-s + 0.328·37-s + 1.29·38-s − 1.71·41-s − 0.457·43-s + 0.301·44-s − 0.737·46-s + 0.583·47-s + 1/7·49-s − 0.970·52-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 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−8.166195220376784091487886220168, −7.22531868709035507083656440499, −6.94848890814954560488195605397, −5.92888425773327956739477407381, −5.17164433623471019617617102230, −4.45141323859402561248630546370, −3.59705674893757768616714541183, −2.65159047596254204642930481562, −1.79404224113914193650767309445, 0,
1.79404224113914193650767309445, 2.65159047596254204642930481562, 3.59705674893757768616714541183, 4.45141323859402561248630546370, 5.17164433623471019617617102230, 5.92888425773327956739477407381, 6.94848890814954560488195605397, 7.22531868709035507083656440499, 8.166195220376784091487886220168