| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s − 14-s + 16-s − 20-s − 22-s + 4·23-s + 25-s − 28-s − 2·29-s − 2·31-s + 32-s + 35-s − 6·37-s − 40-s − 8·41-s − 12·43-s − 44-s + 4·46-s + 6·47-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.267·14-s + 1/4·16-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.188·28-s − 0.371·29-s − 0.359·31-s + 0.176·32-s + 0.169·35-s − 0.986·37-s − 0.158·40-s − 1.24·41-s − 1.82·43-s − 0.150·44-s + 0.589·46-s + 0.875·47-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.28368714736850863244770934380, −7.01888094078790625243594148743, −6.15882837148496197884190866614, −5.34268993699437413784231515090, −4.83805461091601504345378261079, −3.86453670687481216505709131617, −3.32948851620704505026550830580, −2.49617657136681787126926377711, −1.42322882060499839890271457161, 0,
1.42322882060499839890271457161, 2.49617657136681787126926377711, 3.32948851620704505026550830580, 3.86453670687481216505709131617, 4.83805461091601504345378261079, 5.34268993699437413784231515090, 6.15882837148496197884190866614, 7.01888094078790625243594148743, 7.28368714736850863244770934380
